Methods and devices comprising flexible seals, flexible microchannels, or both for modulating or controlling flow and heat

ABSTRACT

Disclosed herein are devices comprising at least one flexible seal, at least one flexible complex seal having at least one closed cavity containing a fluid, or a combination thereof. The devices may comprise at least one immobile and inflexible substrate and at least one mobile and inflexible substrate capable of movement due to the flexible seal, the flexible complex seal, or both. The flexible complex seals comprise at least one closed cavity comprising a fluid, such as a gas or a liquid. As disclosed, the presence or absence of heat will cause the mobile and inflexible substrate to move. The movement will increase or decrease the fluid amount or fluid flow rate in the primary fluid layer. Also disclosed are methods for enhancing the insulating properties of insulating assemblies.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 10/840,303, filed 7 May 2004, pending, which claims the benefitof U.S. Provisional Patent Application No. 60/470,850 filed 16 May 2003,which names Kambiz Vafai and Abdul Rahim A. Khaled as inventors, whichare herein incorporated by reference in their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to thin film channels,microfluidic devices, biosensors, electronic cooling, control of fuelflow prior combustion and insulating assemblies.

2. Description of the Related Art

Thin films are used in a variety of devices, including electrical,electronic, chemical, and biological devices, for modulating orcontrolling flow and heat characteristics in the devices. See e.g. Vafai& Wang (1992) Int. J. Heat Mass Transfer 35:2087-2099, Vafai et al.(1995) ASME J Heat Transfer 117:209-218, Zhu & Vafai (1997) Int. J. HeatMass Transfer 40:2887-2900, and Moon et al. (2000) Int. J. Microcircuitsand Electronic Packaging 23:488-493 for flat heat pipes; Fedorov &Viskanta (2000) Int. J. Heat Mass Transfer 43:399-415, Lee and Vafai(1999) Int. J. Heat Mass Transfer 42:1555-1568, and Vafai & Zhu (1999)Int. J. Heat Mass Transfer 42; 2287-2297 for microchannel heat sinks;Lavrik et al. (2001) Biomedical Microdevices 3(1):35-44, and Xuan &Roetzel (2000) Int. J. Heat Mass Transfer 43:3701-3707 for biosensorsand nanodevices.

For many of these applications, modulation and control of the flow andheat characteristics in the devices is desired. Unfortunately, the priorart methods for modulating and controlling the flow and heat aredifficult or problematic. For example, a two phase flow in amicrochannel is capable of removing maximum heat fluxes generated byelectronic packages, but instability occurs near certain operatingconditions. See Bowers & Mudwar (1994) ASME J. Electronic Packaging116:290-305. Further, the use of porous medium for cooling electronicdevices enhances heat transfer via the increase in the effective surfacearea, but the porous medium results in a substantial increase in thepressure drop inside the thin film. See Huang & Vafai (1993) Int. J.Heat Mass Transfer 36:4019-4032, Huang & Vafai (1994) AIAA J.Thermophysics and Heat Transfer 8:563-573, Huang & Vafai (1994) Int. J.Heat and Fluid Flow 15:48-61, and Hadin (1994) ASME J. Heat Transfer116:465-472.

Therefore, a need still exists for methods of modulating or controllingheat and flow characteristics in thin films.

SUMMARY OF THE INVENTION

The present invention generally relates to thin film channels,microfluidic devices, biosensors, electronic cooling, control of fuelflow prior to combustion and insulating assemblies.

The present invention provides methods to modulate flow and heat in avariety of thermal systems including thin film channels, microfluidics,insulating assemblies, and the like with no need for external cooling orflow controlling devices.

The present invention provides several devices for modulating flow andheat. Several devices provided herein reduce the temperature as thethermal load increases as related to electronic cooling and cooling ofengine applications. Several devices provided herein reduce the flowrate as the thermal load increases which are important to internalcombustion applications where fuel rate needs to be reduced as theengine gets overheated. Several devices provided herein conserve thermalenergy as the temperature increases and to reduce leakage frommicrofluidics. These devices have applications related to thermalinsulations and biosensor devices among others.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are intended to provide further explanation of the invention asclaimed. The accompanying drawings are included to provide a furtherunderstanding of the invention and are incorporated in and constitutepart of this specification, illustrate several embodiments of theinvention, and together with the description serve to explain theprinciples of the invention.

DESCRIPTION OF THE DRAWINGS

This invention is further understood by reference to the drawingswherein:

FIG. 1 shows an insulating assembly comprising the flexible seals of thepresent invention.

FIG. 2 shows primary fluid layer expansion versus its temperature.

FIG. 3 shows the percentage volumetric thermal expansion for theconditions of isobaric expansion and expansion using a linearized modelunder linearly varying pressure.

FIG. 4 shows dimesionless change in the equivalent resistance of thefluid layers for two different fluids.

FIG. 5 shows enhanced insulating properties using xenon and aninsulating assembly using the flexible seals according to the presentinvention.

FIG. 6 shows deteriorated insulating properties using helium and aninsulating assembly using the flexible seals according to the presentinvention.

FIG. 7 shows reduction of thermal losses at large operating temperaturesusing xenon and an insulating assembly using the flexible sealsaccording to the present invention.

FIG. 8 shows deterioration of thermal losses at large operatingtemperatures using helium and an insulating assembly using the flexibleseals according to the present invention.

FIG. 9 shows advanced assemblies with enhanced insulating propertiescomprising the flexible seals according to the present invention.

FIG. 10 shows the schematic diagram for a thin film and the coordinatesystem.

FIG. 11 shows the effects of the fixation parameter on the thin filmthickness.

FIG. 12 shows the effects of the fixation parameter on the fluctation atthe upper substrate.

FIG. 13 shows the effects of the frequency of internal pressurepulsations on the fluctation at the upper substrate.

FIG. 14 shows the effects of the squeezing number on the thin filmthickness.

FIG. 15 shows the effects of the squeezing number on the fluctation atthe upper substrate.

FIG. 16 shows the effects of the phase shift of the internal pressure onthe thin film thickness.

FIG. 17 shows the effects of the thermal squeezing paremeter and thefixation parameter on the mean bulk temperature.

FIG. 18 shows the effects of the thermal squeezing paremeter and thefixation parameter on the average lower substrate temperature.

FIG. 19 shows the effects of the fixation parameter on the Nusseltnumber for constant wall temperature conditions.

FIG. 20 shows the effects of the fixation parameter on the Nusseltnumber for uniform wall heat flux conditions.

FIG. 21 shows the axial development of the Nusselt number vesrus thefixation parameter.

FIG. 22 shows the effects of the frequency of pulsations on the averageheat transfer.

FIG. 23 shows the effects of the frequency of pulsations on the averagelower substrate temperature.

FIG. 24 shows the effects frequency of pulsations on the fluctuation inthe average heat transfer and the average lower substrate temperature.

FIG. 25A is a 3D view of a schematic diagram for a two-layered thin filmsupported by flexible seals and flexible complex seals of the presentinvention.

FIG. 25B shows the front and side views including the main boundaryconditions of the schematic diagram for a two-layered thin filmsupported by flexible seals and flexible complex seals of the presentinvention.

FIG. 26A shows the effects of E* on Ψ_(X=0.5) and dH₁/dτ*(H_(t)=2.0,E₁*=E₂*=E*, F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2,φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=3.0, σ₂=6.0).

FIG. 26B shows the effects of E* on Θ_(AVG) and (θ_(u))_(AVG)(H_(t)=2.0, E₁*=E₂*=E*, F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3,β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=3.0, σ₂=6.0).

FIG. 27A shows the effects of F_(T) on Ψ_(X=0.5) and dH₁/dτ* (H₁=2.0,E₁*=0.3, E₂*=0.003, P_(S1)=1.0, P_(S2)=0.012, β_(p)=0.3, β_(q)=0.2,φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=6.0, σ₂=1.0).

FIG. 27B shows the effects of F_(T) on Θ_(AVG) and (θ_(u))_(AVG)(H_(t)=2.0, E₁*=0.3, E₂*=0.003, P_(S1)=1.0, P_(S2)=0.012, β_(p)=0.3,β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=6.0, σ₂=1.0).

FIG. 28 shows the effects of F_(T) on Nusselt numbers for primary andsecondary flows: (primary flow maintained at a CIF condition, H_(t)=2.0,E₁*=E₂*=E*=0.2, P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2,γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=3.0, σ₂=6.0).

FIG. 29A shows the effects of σ₁ on Ψ_(X=0.5) and dH₁/dτ* (H_(t)=2.0,E₁*=E₂*=E*=0.2, F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2,φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₂=6.0).

FIG. 29B shows the effects of σ₁ on Θ_(AVG) and (θ_(u))_(AVG)(H_(t)=2.0, E₁*=E₂*=E*=0.2, F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3,β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₂=6.0).

FIG. 30A shows the effects of P_(S2) on Ψ_(X=0.5) and dH₁/dτ*(H_(t)=2.0, E₁*=E₂*=E*=0.2, F_(T)=0.15, P_(S1)=1.0, β_(p)=0.3,β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=5.0, σ₂=6.0).

FIG. 30B shows the effects of P_(S2) on Θ_(AVG) and (θ_(u))_(AVG)(H_(t)=2.0, E₁*=E₂*=E*=0.2, F_(T)=0.15, P_(S1)=1.0, β_(p)=0.3,β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=5.0, σ₂=6.0).

FIG. 31 shows the effects of P_(S2) on Nusselt numbers for primary andsecondary flows: (primary flow maintained at a CIF condition, H_(t)=2.0,E₁*=E₂*=E*=0.2, F_(T)=0.15, P_(S1)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2,γ=3.0, γ_(p)=6.0, λ₁=λ₂=0, σ₁=5.0, σ₂=6.0).

FIG. 32A shows the effects of λ₂ on Ψ_(X=0.5) and dH₁/dτ* (primary flowmaintained at a CIP condition, H_(t)=2.0, E₁*=E₂*=E*=0.2, F_(T)=0.15,P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0,λ₁=0, σ₁=3.0, σ₂2=6.0).

FIG. 32B shows the effects of λ₂ on Θ_(AVG) and (θ_(u))_(AVG) (primaryflow maintained at a CIP condition, H_(t)=2.0, E₁*=E₂*=E*=0.2,F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0,γ_(p)=6.0, λ₁=0, σ₁=3.0, σ₂=6.0).

FIG. 33A shows the effects of λ₂ on Ψ_(X=0.5) and dH₁/dτ* (primary flowmaintained at a CIF condition, H_(t)=2.0, E₁*=E₂*=E*=0.2, F_(T)=0.15,P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0,λ₁=0, σ₁=3.0, σ₂=6.0).

FIG. 33B shows the effects of λ₂ on Θ_(AVG) and (θ_(u))_(AVG) (primaryflow maintained at a CIF condition, H_(t)=2.0, E₁*=E₂*=E*=0.2,F_(T)=0.15, P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0,γ_(p)=6.0, λ₁=0, σ₁=3.0, σ₂=6.0).

FIG. 34 shows the effects of γ_(p) on ΔΨ_(X=0.5) and Δ(θ_(u))_(AVG) forthe CIP condition (H_(t)=2.0, E₁*=E₂*=E*=0.3, F_(T)=0.3,P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0, λ₁=λ₂=0,σ₁=3.0, σ₂=6.0).

FIG. 35 shows the effects of H_(t) on Ψ_(X=0.5) and dH₁/dτ* (primaryflow maintained at a CIP condition, E₁*=E₂*=E*=0.2, F_(T)=0.15,P_(S1)=P_(S2)=1.0, β_(p)=0.3, β_(q)=0.2, φ_(p)=π/2, γ=3.0, γ_(p)=6.0,λ₁=λ₂=0, σ₁=3.0, σ₂=6.0).

FIG. 36A is a front view of a schematic diagram for a thin film withflexible complex seal according to the present invention and thecorresponding coordinate system.

FIG. 36B is a side view of a schematic diagram for a thin film withflexible complex seal according to the present invention and thecorresponding coordinate system.

FIG. 36C is a 3D diagram of a schematic diagram for a thin film withflexible complex seal according to the present invention and thecorresponding coordinate system.

FIG. 37A shows the effects of the dimensionless thermal expansionparameter F_(T) on dimensionless thin film thickness H.

FIG. 37B shows the effects of the dimensionless thermal expansionparameter F_(T) on dimensionless average lower substrate temperature(θ_(W))_(AVG).

FIG. 37C shows the effects of the dimensionless thermal expansionparameter F_(T) on dH/dτ.

FIG. 37D shows the effects of the dimensionless thermal expansionparameter F_(T) on exit Nusselt number Nu_(L)

FIG. 38A shows the effects of the dimensionless thermal dispersionparameter λ on dimensionless average lower substrate temperature(θ_(W))_(AVG).

FIG. 38B shows the effects of the dimensionless thermal dispersionparameter λ on dimensionless thickness H.

FIG. 38C shows the effects of the dimensionless thermal dispersionparameter λ on temperature profile.

FIG. 38D shows the effects of the dimensionless thermal dispersionparameter λ on exit Nusselt number Nu_(L)

FIG. 39 shows effects of the dimensionless dispersion parameter λ on thetime variation of the dimensionless thin film thickness dH/dτ.

FIG. 40A shows effects of the thermal squeezing parameter P_(S) and thesqueezing number σ on dimensionless average lower substrate temperature(θ_(W))_(AVG).

FIG. 40B shows effects of the thermal squeezing parameter P_(S) and thesqueezing number σ on dimensionless thin film thickness H.

FIG. 40C shows effects of the thermal squeezing parameter P_(S) and thesqueezing number σ on dH/dτ.

FIG. 41A shows effects of the fixation parameter F_(n) and thedimensionless thermal load amplitude β_(q) on dimensionless averagelower substrate temperature (θ_(W))_(AVG).

FIG. 41B shows effects of the fixation parameter F_(n) and thedimensionless thermal load amplitude β_(q) on dimensionless thin filmthickness H.

FIG. 42 shows effects of the dimensionless thermal expansion parameterF_(T) on the average dimensionless pressure inside the thin filmΠ_(AVG).

FIG. 43A is a schematic diagram of a symmetrical fluidic cell (it has auniform variation in the film thickness under disturbed conditions andcan be used for multi-detection purposes).

FIG. 43B is a schematic diagram of corresponding coordinate systems withleakage illustration.

FIG. 44A shows effects of the dimensionless leakage parameter M_(L) onthe dimensionless thin film thickness H, the film thickness decreaseswith an increase in the leakage.

FIG. 44B shows effects of the dimensionless leakage parameter M_(L) onthe inlet pressure gradient.

FIG. 45 shows effects of the fixation parameter F_(n) on the fluctuationrate at the upper substrate dH/dτ. The fluctuation rate increases as theseal becomes softer.

FIG. 46 shows effects of the squeezing number σ on the fluctuation rateat the upper substrate dH/dτ. The fluctuation rate decreases as theorder of the inlet velocity decreases compared to the axial squeezedvelocity due to pressure pulsations.

FIG. 47A shows the effects of the dimensionless slip parameterβ_(P)/h_(o) on the dimensionless wall slip velocity U_(slip).

FIG. 47B shows the effects of the dimensionless slip parameterβ_(P)/h_(o) on the dimensionless normal velocity V (the dimensionlesstime τ*=3π/2 corresponds to the time at which the fluctuation rate atthe upper substrate is maximum while τ*=11π/6 corresponds to the time atwhich the fluctuation rate at the upper substrate is minimum).

FIG. 48A shows the effects of the power law index n on the dimensionlesswall slip velocity U_(slip).

FIG. 48B shows the effects of the power law index n on the dimensionlessnormal velocity V (the dimensionless time τ*=3π/2 corresponds to thetime at which the fluctuation rate at the upper substrate is maximumwhile τ*=11π/6 corresponds to the time at which the fluctuation rate atthe upper substrate is minimum).

FIG. 49 shows the effects of the dimensionless leakage parameter M_(L)on the average dimensionless lower substrate temperature θ_(W). Thecooling increases with an increase in the leakage rate.

FIG. 50 shows the effects of the fixation parameter F_(n) on the averagedimensionless lower substrate temperature θ_(W). The cooling increasesas the seal becomes softer.

FIG. 51 shows the effects of the squeezing number σ on the averagedimensionless lower substrate temperature θ_(W). The cooling increasesas the order of the inlet velocity increases.

FIG. 52 shows a multi-compartment fluidic cell.

FIG. 53A shows systems with increased cooling capacity as thermal loadincreases utilizing a flexible complex seal according to the presentinvention.

FIG. 53B shows systems with increased cooling capacity as thermal loadincreases utilizing a bimaterial upper substrate.

FIG. 53C illustrates a systems with increased cooling capacity, inaccordance with the disclosed embodiments.

FIG. 54A shows systems with decreased cooling capacity as thermal loadincreases utilizing flexible complex seals according to the presentinvention and two layered thin films.

FIG. 54B shows systems with decreased cooling capacity as thermal loadincreases utilizing a bimaterial upper substrate.

FIG. 54C illustrates systems with decreased cooling capacity, inaccordance with the disclosed embodiments.

FIG. 55A shows an insulating assembly arrangement for low temperatureapplications.

FIG. 55B shows an insulating assembly arrangement for high temperatureapplications.

FIG. 56 shows expected sample results for xenon with and without theflexible seals of the present invention.

FIG. 57 shows a thin film supported by flexible complex seals of thepresent invention with one inlet port and two exit ports.

FIG. 58 illustrates a configuration with primary and secondary layers inassociation with a conductive substrate an an insulative substrate, inaccordance with the disclosed embodiments.

FIG. 59A illustrates a schematic for an cell supported by flexiblecomplex seals.

FIG. 59B is a schematic for an open ended cell supported by flexiblecomplex seals.

FIG. 60A is a front view of a schematic diagram and the coordinatesystem for a single layer flexible microchannel heat sink of the presentinvention.

FIG. 60B is a side view of a schematic diagram and the coordinate systemfor a single layer flexible microchannel heat sink of the presentinvention.

FIG. 61A is a front view of a schematic diagram and the coordinatesystem for a double layered flexible microchannel heat sink of thepresent invention

FIG. 61B is a side view of a schematic diagram and the coordinate systemfor a double layer flexible microchannel heat sink of the presentinvention.

FIG. 62 show effects of the pressure drop

$\left( {{Re}_{o} = {\frac{\rho}{12\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} \right)$on the dimensionless exit mean bulk temperature for a single layerflexible microchannel heat sink.

FIG. 63 shows effects of the pressure drop

$\left( {{Re}_{o} = {\frac{\rho}{12\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} \right)$on the dimensionless average lower plate temperature for a single layerflexible microchannel heat sink.

FIG. 64 shows effects of the pressure drop

$\left( {{Re}_{o} = {\frac{\rho}{12\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} \right)$on the dimensionless average convective heat transfer coefficient for asingle layer flexible microchannel heat sink.

FIG. 65 shows effects of the pressure drop

$\left( {{Re}_{o} = {\frac{\rho}{12\;\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} \right)$on U_(Reo) and U_(F) for a single layer flexible microchannel heat sink.

FIG. 66 shows effects of the fixation parameter on the fully developedheated plate temperature at the exit for a single layer flexiblemicrochannel heat sink.

FIG. 67 shows effects of Prandtl number on the dimensionless averagelower plate temperature for a single layer flexible microchannel heatsink.

FIG. 68 shows effects of Prandtl number on the average convective heattransfer coefficient for a single layer flexible microchannel heat sink.

FIG. 69 shows effects of the fixation parameter on the mean bulktemperature inside the double layered flexible microchannel heat sink

FIG. 70 shows effects of the pressure drop

$\left( {{Re}_{o} = {\frac{\rho}{12\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} \right)$on κ_(m) and κ_(W)

FIG. 71 shows effects of the pressure drop

$\left( {\left( {Re}_{o} \right)_{DL} = {\frac{\rho}{12\;\mu^{2}}\frac{\left( {\Delta\; p} \right)_{DL}}{B}H_{o}^{3}}} \right)$on the pressure drop ratio and the friction force ratio between singleand double layered flexible microchannel heat sinks.

FIG. 72 shows effects of the delivered coolant mass flow rate on theaverage heated plate temperature for both single and double layeredflexible microchannel heat sinks.

FIG. 73 is a schematic diagram and the coordinate system.

FIG. 74 shows different arrangements for the thermal dispersion region:(a) central arrangement, and (b) boundary arrangement.

FIG. 75 shows effects of the thermal dispersion parameter E_(o) and thedimensionless thickness Λ on the Nusselt number at thermally fullydeveloped conditions for the central arrangement (the number of thedispersive elements is the same for each arrangement).

FIG. 76 shows effects of the thermal dispersion parameter E_(o) and thedimensionless thickness Λ on the Nusselt number at thermally fullydeveloped conditions for the boundary arrangement (the number of thedispersive elements is the same for each arrangement).

FIG. 77 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the Nusselt number at the exit for centralarrangement (the number of the dispersive elements is the same for eacharrangement).

FIG. 78 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the average dimensionless plate temperatureθ_(W) for central arrangement (the number of the dispersive elements isthe same for each arrangement, Pe_(f)=670).

FIG. 79 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the average dimensionless plate temperatureθ_(W) for central arrangement (the number of the dispersive elements isthe same for each arrangement, Pe_(f)=1340).

FIG. 80 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the Nusselt number at the exit for theboundary arrangement (the number of the dispersive elements is the samefor each arrangement).

FIG. 81 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the average dimensionless plate temperatureθ_(W) for boundary arrangement (the number of the dispersive elements isthe same for each arrangement, Pe_(f)=670).

FIG. 82 shows effects of the dispersion coefficient C* and thedimensionless thickness Λ on the average dimensionless plate temperatureθ_(W) for boundary arrangement (the number of the dispersive elements isthe same for each arrangement, Pe_(f)=340).

FIG. 83 shows effects of D_(e) on the volume fraction distribution ofthe dispersive element (the number of the dispersive elements is thesame for each distribution).

FIG. 84 shows effects of D_(c) on the volume fraction distribution ofthe dispersive elements (the number of the dispersive elements is thesame for each distribution).

FIG. 85 shows effects of D_(e) on the fully developed value for theNusselt number (exponential distribution, the number of the dispersiveelements is the same for each distribution).

FIG. 86 shows effects of D_(c) on the fully developed value for theNusselt number (parabolic distribution, the number of the dispersiveelements is the same for each distribution).

FIG. 87 is a graph that shows that the excess in Nusselt number κ isalways greater than one for the boundary arrangement while it is greaterthan one for the exponential distribution when the velocity is uniform.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides methods for modulating or controllingheat and flow characteristics in a variety of devices. In particular,the present invention provides flexible seals for modulating orcontrolling heat and flow characteristics in devices comprising thinfilms, such as thin film channels, microchannels, microfluidics and thelike. The present invention also provides a method to control heat andflow inside other thermal systems, such as insulating assemblies andfuel flow passages. As used herein, a “flexible seal” refers to amaterial that can be deformed significantly according to the load actingupon it. Examples of these materials include elastmors, polymers,natural rubber, closed rubber cell foams, and the like. In someembodiments, the present invention provides flexible complex seals formodulating or controlling heat and flow characteristics in devicescomprising thin films, such as microchannels and microfluidics. As usedherein, a “flexible complex seal” refers to a flexible seal comprisingat least one closed cavity of stagnant fluid. In preferred embodiments,the stagnant fluid has at least one point of contact with the heatedsurface of the device. In preferred embodiments, the stagnant fluid hasa large value of the volumetric thermal coefficient. As used herein, a“fluid” refers to a continuous amorphous substance that tends to flowand to conform to the outline of a container, such as a liquid or a gas,and may be used in accordance with the present invention. As usedherein, “stagnant fluid” refers to a fluid that is not circulating orflowing and in preferred embodiments of the present invention, thestagnant fluid is surrounded by a flexible seal of the present inventionand/or the surfaces of a device such that the average translationalvelocity of the fluid is zero.

As used herein, “primary fluid” refers to the fluid that the devices ofthe present invention control or modulate its flow rate or itstemperature. As used herein, “secondary fluid” refers to an auxiliaryfluid utilized in the present invention to achieve additional controland modulation features for the primary fluid flow rate and temperature.As provided herein, the stagnant fluid in the complex flexible seals canhave characteristics that are the same as or different from thecharacteristics of the primary fluid, the secondary fluid, or both. Asused herein, “biofluid” refers to the fluid that contains at least onespecies of a biological substance that needs to be measured. As providedherein, the primary fluid can be a biofluid.

The flexible seals and flexible complex seals of the present inventionare typically found between a first substrate and a second substrate ofa thin film or other thermal systems such as the insulating assemblies.As used herein, “substrate” includes plates which may be inflexible orflexible according to part 6 herein below. In some preferredembodiments, the elastic modulus for the seals of the present invention,the ratio of the applied stress on the seal to the induced strain, rangefrom about 10³N/m² to about 10⁷N/m². The seals of the present inventionmay comprise at least one closed cavity of a fluid such as air or thelike in order to minimize their effective elastic modulus. Thedeformation of the flexible seals of the present invention can be guidedby special guiders to attain maximum or desired deformations. Inpreferred embodiments, the flexible seals comprise differentcross-sectional geometries, such as circular cross-section, rectangularcross-section and the like. As used herein, “thin films” include fluidicdevices that have the thickness of their fluidic layers of an order ofabout a millimeter or less such as, microchannels and microfluidicdevices. Thin films comprise at least two substrates, lower and uppersubstrates, and at least one fluidic layer. As used herein, an“insulating assembly” means an assembly of at least two insulatingsubstrates and at least one fluid layer placed consecutively in series.

The flexible seals and flexible complex seals of the present inventionare typically found between a first substrate and a second substrate ofa thin film or other thermal systems such as the insulating assemblies.As used herein, “substrate” includes plates which may be inflexible orflexible according to part 6 herein below. In some preferredembodiments, the elastic modulus for the seals of the present invention,the ratio of the applied stress on the seal to the induced strain, rangefrom about 10³N/m² to about 10⁷N/m². The seals of the present inventionmay comprise at least one closed cavity of a fluid such as air or thelike in order to minimize their effective elastic modulus. Thedeformation of the flexible seals of the present invention can be guidedby special guiders to attain maximum or desired deformations. Inpreferred embodiments, the flexible seals comprise differentcross-sectional geometries, such as circular cross-section, rectangularcross-section and the like. As used herein, “thin films” include fluidicdevices that have the thickness of their fluidic layers of an order ofabout a millimeter or less such as, microchannels and microfluidicdevices. Thin films comprise at least two substrates, lower and uppersubstrates, and at least one fluidic layer. As used herein, an“insulating assembly” means an assembly of at least two insulatingsubstrates and at least one fluid layer placed consecutively in series.

As disclosed herein, modulating the thermal characteristics of a devicemay be conducted by modifying the thin film thickness, the thermal load,the flow rate, or a combination thereof. For example, additional coolingcan be achieved if the thin film thickness is allowed to increase by anincrease in the thermal load, pressure gradient or both which will causethe coolant flow rate to increase. As provided herein, the enhancementin the cooling due to the flexible complex seals used is substantial atlarger thermal loads for stagnant liquids while this enhancement is muchlarger at lower temperatures for stagnant fluids, especially idealgases. This is because the volumetric thermal expansion coefficientincreases for liquids and decreases for gases as the temperatureincreases. Moreover, the enhancement in the cooling due to flexibleseals is substantial at larger pressure gradients for single layeredthin films while it is significant for double layered thin films atlower pressure gradients.

Khaled and Vafai analyzed the enhancement in the heat transfer insidethin films supported by flexible complex seals. See Khaled & Vafai(2003) ASME J. of Heat Transfer 125:916-925, which is hereinincorporated by reference. Specifically, the applied thermal load wasconsidered to vary periodically with time in order to investigate thebehavior of expandable thin film systems in the presence of a noise inthe applied thermal load. As provided herein, a noticeable enhancementin the cooling capacity can be achieved for large thermal loadsespecially in cooling of high flux electronic components (q≈700 kW/m²)since they produce elevated working temperatures. Also, the generatedsqueezing effects at the mobile and inflexible substrate can beminimized when nanofluids are employed in the coolant flow. As usedherein, “nanofluids” are mixtures of a working fluid, such as water, andsuspended ultrafine particles in the fluid such as copper, aluminum, orthe like with diameters of an order of about the nanometer range. SeeEastman et al. (2001) Applied Physics Letters 78: 718-720, which isherein incorporated by reference.

The flexible seals, flexible complex seals, or both of the presentinvention may be used in two-layered thin films in order to regulate theflow rate of the primary fluid layer such that excessive heating in thesecondary fluid layer results in a reduction in the primary fluid flowrate. For example, the flexible seals, flexible complex seals, or bothof the present invention may be applied in the internal combustionindustry where the fuel flow rate should be reduced as the engine getsoverheated. In this example, the primary fluid flow is the fuel flowwhile the secondary fluid flow can be either flow of combustionproducts, flow of engine coolant or flow of any other auxiliary fluid.The flexible seals, flexible complex seals, or both of the presentinvention may be used to modulate or control exit thermal conditions indevices comprising two-layered thin films. For example, the flexibleseals, flexible complex seals, or both of the present invention may beused to minimize bimaterial effects of various biosensors, includingmicrocantilever based biosensors, which are sensitive to flowtemperatures. See Fritz et al. (2000) Science 288:316-318, which isherein incorporated by reference. In this example, the primary fluidflow is flow of a biofluid while the secondary fluid flow can be eitherflow of the external surrounding fluid or flow of any auxiliary fluid.

As provided herein, thin films comprising flexible seals, flexiblecomplex seals, or both are modeled and designed in order to alleviatethe thermal load or modulate the flow. These systems according to thepresent invention provide noticeable control of the flow rate, reducethermal gradients within the primary fluid layer at relatively largeexternal thermal loads, and minimize fluctuation at the mobile andinflexible substrate in the presence of nanofluids.

1. Control of Insulating Properties Using Flexible Seals

As disclosed herein, the present invention provides a method formodulating or controlling the insulating properties of a device, aninsulating assembly having insulating substrates separated by fluidlayers and flexible seals. The fluid layers were supported by flexibleseals in order to allow for volumetric thermal expansion of the primaryfluid layers while the secondary fluid layers are vented to theatmosphere such that the secondary fluid pressure remains constant. Thevolumetric thermal expansion of the primary fluid layers within theinsulating assembly were determined taking into consideration thevariation in the fluid pressure due to the elastic behavior of thesupporting flexible seals. The volumetric thermal expansion of theprimary fluid layers was correlated to the increase in the equivalentthermal resistance of the fluid layers. The volumetric thermal expansionof the primary fluid was found to approach its isobaric condition valueas the primary fluid layer thickness decreases. Also, the insulatingproperties were found to be enhanced when the primary fluid had aminimum thermal conductivity and when relatively high temperatures wereexperienced. The insulating properties deteriorate at large temperatureswhen the primary fluid has a relatively large thermal conductivity.

The following Table 1 provides the various symbols and meanings used inthis section:

TABLE 1 A_(S) surface area of the intermediate insulating substrateC_(F) volumetric thermal expansion efficiency h_(c) convective heattransfer coefficent at the upper surface h_(o) reference thickness ofthe primary fluid layer K* stiffness of the supporting seal k_(ins)thermal conductivity of insulating substrates k₁ thermal conductivity ofthe primary fluid k₂ thermal conductivity of the secondary fluid m₁ massof the primary fluid p_(atm) pressure of the surrounding q heat flux R₁primary fluid layer fluid constant R_(th) thermal resistance of thefluid layers R_(tho) orginal thermal resistance of the fluid layers Taverage temperature of the primary fluid layer T₁ temperature at thelower surface of the primary fluid layer T_(o) orginal primary fluidtemperature T_(e) temperature of the upper surface facing of thesurroundings Δh₁ expansion of the primary fluid layer η_(R)dimensionless increase in the resistance of the fluid layers

Generally, thermal losses increase at large working temperatures. Thepresent invention provides a device that has desirable insulativeattributes even at high working temperatures. That is, the presentinvention better conserves thermal energy especially at hightemperatures as compared to similar devices that do not compriseflexible seals. An example of a device of the present invention is shownin FIG. 1. The device shown in FIG. 1 comprises the following frombottom to top: (1) a heated substrate (generally due to contact with orproximity to a heat source), (2) a first layer of fluid that has a verylow thermal conductivity such as xenon (the primary fluid layer), (3) athin layer of an insulating substrate, (4) a secondary fluid layercomprising a second fluid that has a lower thermal conductivity like air(needs to be larger than that of the first layer and is open to theoutside environment), and (5) a top insulating substrate. The first andthe second fluid layers along with the intermediate insulating substrateare connected together by flexible seals. Both the heated substrate andthe upper insulating substrate are fixed (immobile and inflexiblesubstrates) while the intermediate insulating substrate is capable ofmoving as it is supported by flexible complex seals (mobile andinflexible substrate). In preferred embodiments, the flexible seals aremade of a material resistant to melting at high temperatures. In orderto avoid melting the seals at high temperatures, ordinary homogenousflexible seals may be replaced with flexible complex seals, a flexibleseal comprising at least one closed cavity containing a fluid, such as agas.

1A. Operational Principle

When the operating temperature (high temperature source) increases, theaverage fluid temperature of the primary fluid layer increases.Accordingly, the volume of the primary fluid layer expands accompaniedby a shrinkage in the secondary fluid layer. As such, an increase in theequivalent thermal resistance of the insulating assembly can be attainedas long as the thermal conductivity of the primary fluid layer issmaller than that for the secondary fluid layer. Preferably, the heatedsubstrate has a relatively small thickness and a relatively largethermal conductivity so that the thermal expansion of the primary fluidlayer is maximized.

1B. Volumetric Expansion in the Primary Fluid Layer

Forces on elastic materials, such as seals, are usually proportional tothe elongation of this material. See R. L. Norton (1998) MACHINE DESIGN:AN INTEGRATED APPROACH Prentice-Hall, NJ, which is herein incorporatedby reference. Accordingly, a force balance on the intermediateinsulating substrate results as provided in Equation 1 as follows:

$\begin{matrix}{{\frac{m_{1}R_{1}T}{A_{S}\left( {h_{o} + {\Delta\; h_{1}}} \right)} - p_{atm}} = {\frac{K^{*}}{A_{S}}\Delta\; h_{1}}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$wherein

T is the average temperature of the primary fluid layer

K* is the stiffness of the supporting seals

A_(s) is the surface area of the intermediate insulating substrate.

h_(o) is the reference thickness of the primary fluid layer

Δh₁ is the corresponding expansion in the primary fluid layer thickness

m₁ is the mass of the primary fluid

R₁ is the primary fluid constant

The first term on the left hand side of Equation 1 represents thepressure inside the primary fluid layer. The reference thickness h_(o)corresponds to the thickness of the primary fluid layer when the primaryfluid pressure is equal to the atmospheric pressure. Equation 1 can besolved for Δh₁ and the expansion is found to be:

$\begin{matrix}{\frac{\Delta\; h_{1}}{h_{o}} = {C_{1}\left( {\sqrt{\frac{C_{2}}{C_{1}^{2}} + 1} - 1} \right)}} & {{Eq}.\mspace{14mu} 2} \\{where} & \mspace{11mu} \\{C_{1} = \frac{{\left( {p_{atm}A_{S}} \right)/\left( {K^{*}h_{o}} \right)} + 1}{2}} & {{Eq}.\mspace{14mu} 3} \\{and} & \; \\{C_{2} = {{\left( {m_{1}R_{1}T} \right)/\left( {K^{*}h_{o}^{2}} \right)} - {\left( {p_{atm}A_{S}} \right)/\left( {K^{*}h_{o}} \right)}}} & {{Eq}.\mspace{14mu} 4}\end{matrix}$

In order to maximize the expansion in the primary fluid layer which inturn results in better insulating properties, i.e. increased effectivethermal resistance of the insulating assembly, the parameter C₂ needs tobe maximized. This can be accomplished by considering minimum values ofK*h_(o) while the following relationship provided in Equation 5 ispreferred to be satisfied:

$\begin{matrix}{\frac{m_{1}R_{1}T}{p_{atm}A_{S}h_{o}}\operatorname{>>}1} & {{Eq}.\mspace{14mu} 5}\end{matrix}$

The following parameters were considered for studying the flexible sealsof the present invention: K*=48000 N/m, A_(S)=0.0036 m² and p_(atm)=0.1Mpa. The parameter m₁R₁ was evaluated at the reference condition whenthe primary fluid pressure was equal to the atmospheric pressure. Thiscondition which causes the expansion to be zero in Equation 1 wasassumed to be at T=T_(o)=283 K and h_(o)=0.004 m. This leads tom₁R₁=5.088×10⁻³ J/K. Accordingly, the relation between the volumetricthermal expansion of the primary fluid layer and its average temperatureis illustrated in FIG. 2.

Equation 2 reduces to the following linearized model for relatively lowvolumetric thermal expansion levels

$\left( {\frac{\Delta\; h_{1}}{h_{o}} < 0.2} \right)\text{:}$

$\begin{matrix}{{\frac{\Delta\; h_{1}}{h_{o}} \approx {{0.5\frac{C_{2}}{C_{1}}} + {O\left( {\Delta\; h_{1}^{2}} \right)}}} = {\frac{T - T_{o}}{T_{o} + \frac{K^{*}h_{o}^{2}}{m_{1}R_{1}}} + {O\left( {\Delta\; h_{1}^{2}} \right)}}} & {{Eq}.\mspace{14mu} 6}\end{matrix}$where T_(o) is the average temperature of the primary fluid layer at thereference condition. The reference condition corresponds to thecondition that produces a zero net force on the seals. That is, thermalexpansion is zero when the primary fluid layer is kept at T_(o). At thiscondition, the primary fluid layer thickness is h_(o). The relativevolumetric thermal expansion, Δh_(l)/h_(o), approximated by Equation 6is similar to that for isobaric expansion with the average primary fluidtemperature being increased by the parameter

$\frac{K^{*}h_{o}^{2}}{m_{1}R_{1}}.$This parameter is denoted as ΔT_(o).

The error associated with Equation 6 is further reduced if

$\frac{m_{1}R_{1}T_{o}}{K^{*}h_{o}^{2}} > 1.$The latter inequality means that the insulating system exhibitsrelatively large volumetric thermal expansion by having a small increasein the primary fluid pressure due to the elastic behavior of theflexible seal. FIG. 3 illustrates the difference between the relativevolumetric expansion expressed by Equation 6 and that obtained when theexpansion is at a constant pressure. FIG. 3 shows that isobaricconditions provide favorable volumetric thermal expansion when comparedto volumetric thermal expansion under linearly varying pressure as whenflexible seals are present.

The efficiency of the volumetric thermal expansion C_(F) of the primaryfluid layer is defined as the ratio of the expansion in the primaryfluid layer when the flexible seal is present to the expansion whenunder constant pressure as expressed in the following Equation 7:

$\begin{matrix}{C_{F} = \frac{\Delta\; h_{1}}{\left( {\Delta\; h_{1}} \right)_{Isobaric}}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$where (Δh₁)_(Isobaric)/h_(o)=(T−T_(o))/T_(o). For the linearized modelshown in Equation 6, the efficiency C_(F) will be:

$\begin{matrix}{C_{F} \cong \frac{T_{o}}{T_{o} + {\Delta\; T_{o}}}} & {{Eq}.\mspace{14mu} 8}\end{matrix}$

According to Equation 8, the values of C_(F) which approaches unity asΔT_(o) decreases are provided for various ΔT_(o) in Table 2 as follows:

TABLE 2 Volumetric thermal expansion efficiency C_(F) of the primaryfluid layer versus ΔT_(o) ΔT_(o) (K) C_(F) (T_(o) = 283 K) 10 0.966 500.850 100 0.739 150.88 0.6521C. Equivalent Thermal Resistance of Fluid Layers

The equivalent thermal resistance of the fluid layers during volumetricthermal expansion is given by the following Equation 9:

$\begin{matrix}{R_{th} = {\frac{h_{o}}{k_{1}} + \frac{h_{o}}{k_{2}} + {\Delta\;{h_{1}\left( {\frac{1}{k_{1}} - \frac{1}{k_{2}}} \right)}}}} & {{Eq}.\mspace{14mu} 9}\end{matrix}$where

k₁ is the thermal conductivity of the primary fluid

k₂ is the thermal conductivity of the secondary fluid.

Both fluid layers are assumed to have a similar thickness prior tothermal expansion equal to h_(o). Based on Equation 1 and Equation 3,the increase in the equivalent thermal resistance ΔR_(th), the thirdpart on the right of Equation 9, was correlated to the relativeexpansion in the primary fluid layer according to the following Equation10:

$\begin{matrix}{{\eta_{R} \equiv \frac{\Delta\; R_{th}}{R_{tho}}} = {\frac{\Delta\; h_{1}}{h_{o}}\frac{\left( {k_{2} - k_{1}} \right)}{\left( {k_{1} + k_{2}} \right)}}} & {{Eq}.\mspace{14mu} 10}\end{matrix}$where R_(tho) is the equivalent thermal resistance of both layers priorto thermal expansion.

The parameter R_(tho) is the sum of the first two terms on the right ofEquation 9. When the parameter η_(R) is positive, the thermal resistanceof the insulating assembly increases while it decreases as it becomesnegative. Therefore, R_(tho) represents the dimensionless increase inthe thermal resistance. Various properties of different gases areprovided in the following Table 3:

TABLE 3 Various Properties of Proposed Different Gases at T = 373 K andp = 1 atm Primary fluid k(W/mK) ρ(kg/m³) R(J/kg K) (k_(air) −k)/(k_(air) + k) Xenon 0.0068 4.3 64.05 0.609 Krypton 0.011 2.75 99.780.4359 Helium 0.181 0.13 2077 −0.732 Neon 0.0556 0.66 412.1 −0.33 Argon0.0212 1.3 209 0.138 Air 0.028 1.2 287 0

According to Table 3, xenon can be used to enhance the insulatingproperties while helium is preferable to deteriorate the insulatingproperties especially at large operating temperatures as can be noticedfrom the last column in Table 3.

FIG. 4 shows the dimensionless increase in the fluid layers equivalentthermal resistance when the primary fluid layer is charged with xenon orhelium while the secondary fluid layer is open to the atmosphere.Charging the primary fluid layer with xenon can provide about a 20percent increase in the effective thermal resistance of the fluid layerswith an increase of the primary fluid layer temperature by about 165 K.However, helium can produce a deterioration in the insulating propertiesby about 25 percent with about a 165 K increase in the primary fluidlayer temperature.

1D. Heat Transfer Analysis

In the following analysis, the temperature at the lower side of theprimary fluid layer was assumed to be kept under T₁. See FIG. 1. Theinsulating substrates were assumed to have equal thicknesses and thermalconductivities which were equal to the reference thickness for theprimary fluid layer h_(o) and k_(ins), respectively. Accordingly, thethermal energy balance on the insulating assembly shown in FIG. 1reveals the following relation for the temperature at the surface of thelower temperature side T_(e) and the heat transfer q, respectively:

$\begin{matrix}{T_{e} = {\frac{\left( {T_{1} - T_{\infty}} \right)/h_{c}}{\left( {\frac{1}{h_{c}} + \frac{2h_{o}}{k_{ins}} + {R_{tho}\left( {1 + \frac{\Delta\; R_{th}}{R_{tho}}} \right)}} \right)} + T_{\infty}}} & {{Eq}.\mspace{14mu} 11} \\{q = \frac{\left( {T_{1} - T_{\infty}} \right)}{\left( {\frac{1}{h_{c}} + \frac{2h_{o}}{k_{ins}} + {R_{tho}\left( {1 + \frac{\Delta\; R_{th}}{R_{tho}}} \right)}} \right)}} & {{Eq}.\mspace{14mu} 12}\end{matrix}$where

h_(c) is the convective heat transfer coefficient at the lowertemperature side

T_(∞) is the temperature of environment facing the lower temperatureside

The surface area of the insulating assembly that faces the seal isrelatively small. Therefore, the heat transfer through the seal portionis neglected in Equation 11 and Equation 12. For the previous examplealong with h_(c)=5 W/m²K, T_(∞)=275 K and k_(ins)=0.04 W/mK, thetemperature T_(e) as a function of T₁ is illustrated in FIG. 5 and FIG.6, respectively. These figures also compare the temperature T_(e) forthe case when the thermal expansion is encountered due to the presenceof flexible seals with the case where thermal expansion is not present(both fluid layer thicknesses are equal to h_(o) for all values of T₁).FIG. 5 shows that insulating properties are enhanced when xenon andflexible seals are used and that T_(e) for this case is departing awaydown from the values corresponding to the case where the thermalexpansion is not present. Also, this figure shows that the departurerates compared to the case where the thermal expansion is not present,increase as the temperature levels increase.

FIG. 6 shows that insulating properties are deteriorated when helium andflexible seals are used. As shown in FIG. 6 the departure of T_(e) forthis case from the results corresponding to the case with no thermalexpansion is in the direction of an increase in T_(e). Thus, insulatingproperties are deteriorated at larger rates when helium and flexibleseals are used especially at large operating temperatures. The thermalexpansion of the primary fluid layer was computed at its averagetemperature. As such, an iterative procedure was implemented ingenerating FIG. 5 and FIG. 6 so that the obtained temperatures producethe employed thermal expansion of the primary fluid layer. Also, thevolumetric thermal expansion that were used to develop FIG. 5 and FIG. 6were evaluated from Equation 2.

FIG. 7 shows a comparison of heat flux of the insulating assembly withxenon as the primary fluid under the following two conditions: (1) inthe presence of flexible seals, and (2) when thermal expansion is notpresent and the thickness of the fluid layers is h_(o) at all workingtemperatures. FIG. 7 shows a reduction in the heat flux when flexibleseals are introduced. FIG. 7 also shows that the reduction rate in theheat flux increases as the working temperatures increase indicatingbetter insulating characteristics are achieved when flexible seals areused to support the primary fluid layer while the secondary fluid layeris vented. On the other hand, an increase in the heat flux is attainedwhen flexible seals are used to support a fluid layer comprising a fluidwith relatively large thermal conductivity, such as helium, as shown inFIG. 8.

1E. Simplified Correlation

For the insulating assembly shown in FIG. 1, heat transfer can beexpressed by the following Equation 13a:

$\begin{matrix}{q = \frac{\left( {T_{1} - T_{e}} \right)}{\mspace{14mu}\left( {{\sum\limits_{i = 1}^{2}\frac{\left( h_{ins} \right)_{i}}{\left( k_{ins} \right)_{i}}} + {\left( {\frac{h_{o\; 1}}{k_{1}} + \frac{h_{o\; 2}}{k_{2}}} \right)\;\left( {1 + {\frac{\left( {k_{2} - k_{1}} \right)}{\left( {k_{2} + {h_{o\; 2}{k_{1}/h_{o\; 1}}}} \right)}\left( \frac{\Delta\; h_{1}}{h_{o\; 1}} \right)}} \right)}} \right)\mspace{11mu}}} & {{{Eq}.\mspace{14mu} 13}a}\end{matrix}$where Δh₁/h_(o1) can be shown to be equal to the following Equation 13b:

$\begin{matrix}{\frac{\Delta\; h_{1}}{h_{o\; 1}} = {\left( \frac{T_{o} + {\Delta\; T_{o}}}{2\;\Delta\; T_{o}} \right)\left\lbrack {\sqrt{\frac{4\left( {T_{1}^{*} - T_{o}} \right)\Delta\; T_{o}}{\left( {T_{o} + {\Delta\; T_{o}}} \right)^{2}} + 1} - 1} \right\rbrack}} & {{{Eq}.\mspace{14mu} 13}b}\end{matrix}$where

h_(o1) is the reference primary fluid layer thickness

h_(o2) is the reference secondary fluid layer thickness

(h_(ins))_(i) is the thickness of the i^(th) insulating substrate

(k_(ins))_(i) is the thermal conductivity of the i^(th) insulatingsubstrate

T_(o) is the primary fluid layer temperature that causes the primaryfluid pressure to be equal to the atmospheric pressure

T₁* represents the average primary fluid layer temperature

The parameter T₁* can be measured experimentally or determinedtheoretically using an iterative scheme. Equation 13a is based on theassumption that the heat transfer through the flexible seals isnegligible when compared to the total heat transferred through theinsulating assembly.

The solution of Equation 13a and Equation 13b can be used to producepertinent engineering correlations. For example, percentage differencebetween the heat flux including thermal expansion effects and the heatflux at reference condition, q_(ref), where thermal expansion isignored, and correlated to T₁, T_(e), T_(o), k₁ and ΔT_(o). The obtainedfamily of correlations has the following functional form:

$\begin{matrix}{{{\frac{\left( {q_{ref} - q} \right)}{q_{ref}} \times 100}\%} = {\quad{\left\lbrack {a - {b\left( T_{o} \right)} - {c\left( {\Delta\; T_{o}} \right)} - {d\left( k_{1} \right)} + {e\left( {T_{e}T_{o}\Delta\; T_{o}k_{1}} \right)}} \right\rbrack\left( {T_{1} - T_{o}} \right)^{m}\left( \frac{T_{e}}{270} \right)^{n}}}} & {{Eq}.\mspace{14mu} 14}\end{matrix}$where a, b, c, d, e, m, n and the correlation coefficient R² fordifferent values of h_(o1) are listed in Table 4 as follows:

TABLE 4 Coefficients of Equation 14 for different h_(o1) h_(o1)(m)Coefficients R² 0.004 a = 0.559, b = 1.08 × 10⁻³, c = 5.14 × 10⁻⁴, 0.980d = 11.572, e = 2.74 × 10⁻⁷, m = 0.850, n = 0.1.789 0.006 a = 0.591, b =1.17 × 10⁻³, c = 5.26 × 10⁻⁴, 0.983 d = 11.399, e = 2.71 × 10⁻⁷, m =0..847, n = 1.880 0.008 a = 0.610, b = 1.23 × 10⁻³, c = 5.32 × 10⁻⁴,0.984 d = 11.295, e = 2.69 × 10⁻⁷, m = 0.845, n = 1.934This correlation was obtained over the following range of parametervariations: 310<T₁<400 K, 270<T_(o)<290 K, 50<ΔT_(o)<150 K,270<T_(e)<300 K, 0.001<k₁<0.017 W/m K, h_(o2)=h_(o1),

${\frac{\left( h_{ins} \right)_{i}}{\left( k_{ins} \right)_{i}} = {0.2\mspace{14mu} m^{2}{K/W}}},$and k₂=0.028 W/m K.1F. Examples of Insulating Assemblies with Maximum Enhanced InsulatingProperties

FIG. 9 shows a more advanced insulating assembly comprising an array ofprimary and secondary fluid layers supported by flexible seals. Thesecondary fluid layers are vented to the external atmosphere in order toprovide maximum volumetric thermal expansion of primary fluid layers.Accordingly, the insulating properties are enhanced for the assemblyprovided that the primary fluid possesses relatively lower thermalconductivity than the secondary fluid which is the air. The insulatingassembly of FIG. 9A shows the frame of the insulating assembly supportedby a flexible seal, thereby allowing for additional volumetric thermalexpansion for the primary fluids, thereby resulting in furtherenhancements of insulating properties, an increase in the effectivethermal resistance of the assembly. In an alternative embodiment, softelastic balloons having minimized stiffness and containing fluids withminimized thermal conductivities within the secondary fluid layer may beused and placed in a vented layer as shown in FIG. 9B and FIG. 9C. Inthis arrangement the primary fluid layer is eliminated and is suitablefor lower heat flux applications. The degree of enhancements in theinsulating properties of the insulating assemblies of the presentinvention are governed by the temperature levels that the flexible sealscan sustain before melting. Thus, flexible seals having high meltingpoints are preferably used for insulating assemblies for hightemperature applications. The compositions and thus the melting pointsof the flexible seals of the present invention suitable for desiredtemperature conditions may be readily selected by one skilled in the artusing known methods.

2. Flow and Heat Transfer Inside Thin Films Supported by Flexible Sealsin the Presence of Internal and External Pressure Pulsations

As provided herein, the effects of both external squeezing and internalpressure pulsations were studied on flow and heat transfer insidenon-isothermal and incompressible thin films supported by flexibleseals. The laminar governing equations were non-dimensionalized andreduced to simpler forms. The upper substrate (mobile and inflexiblesubstrate) displacement was related to the internal pressure through theelastic behavior of the supporting seals. The following parameters:squeezing number, squeezing frequency, frequency of pulsations, fixationnumber (for the seal) and the thermal squeezing parameter are the maincontrolling parameters. Accordingly, their influences on flow and heattransfer inside disturbed thin films were determined and analyzed. Asprovided herein, an increase in the fixation number results in morecooling and a decrease in the average temperature values of the primaryfluid layer. Also, an increase in the squeezing number decreases theturbulence level at the upper substrate. Furthermore, fluctuations inthe heat transfer and the fluid temperatures may be maximized atrelatively lower frequency of internal pressure pulsations.

The following Table 5 provides the various symbols and meanings used inthis section:

TABLE 5 B Thin film length c_(p) specific heat of the fluid d_(s)effective diameter of the seal E modulus of elasticity for the seal'smaterial F_(n) fixation number H, h, h_(o) dimensionless, dimensionaland reference thin film thickness h_(c) convective heat transfercoefficient k thermal conductivity of the fluid Nu_(L), Nu_(U) lower andupper substrates Nusselt numbers P_(S) thermal squeezing parameter pfluid pressure q reference heat flux at the lower substrate for UHF T,T₁ temperature in fluid and the inlet temperature T₂ temperature at thelower and the upper substrates for CWT t time V_(o) reference axialvelocity U, u dimensionless and dimensional axial velocities V, vdimensionless and dimensional normal velocities X, x dimensionless anddimensional axial coordinates Y, y dimensionless and dimensional normalcoordinates α thermal diffusivity for the fluid β, β_(p) dimensionlesssqueezing motion and pressure pulsation amplitudes ε perturbationparameter γ, γ_(p) dimensionless squeezing motion and pressure pulsationfrequencies μ dynamic viscosity of the fluid θ, θ_(m) dimensionlesstemperature and dimensionless mean bulk temperature θ_(W) dimensionlesstemperature at the lower substrate (UHF) ρ density of the fluid τdimensionless time σ squeezing number ω reciprocal of a reference time(reference squeezing frequency) η variable transformation for thedimensionless Y-coordinate Θ dimensionless heat transfer parameter (CWT)Π dimensionless pressure Π_(i), Π_(o) dimensionless inlet pressure anddimensionless mean pressure

In certain thin film applications, external disturbances, such asunbalances in rotating machines or pulsations in external ambientpressures due to many disturbances, can result in an oscillatory motionat the upper substrate boundary. In addition to external disturbances,internal pressure pulsations such as irregularities in the pumpingprocess, can produce similar oscillatory motion. Even small disturbanceson the substrates of the thin film can have a substantial impact on thecooling process as the thickness of thin films is very small. Thesedisturbances are even more pronounced if the thin film is supported byflexible seals. Accordingly, the dynamics and thermal characterizationof thin films will be altered.

The chambers for chemical and biological detection systems such asfluidic cells for chemical or biological microcantilever probes areexamples of thin films. See Lavrik et al. (2001) Biomedical Devices3(12):35-44, which is herein incorporated by reference. Small turbulencelevels that can be introduced into these cells by either flow pulsatingat the inlet or external noise that may be present at the boundarieswhich result in a vibrating boundary can produce flow instabilitiesinside the fluidic cells. These disturbances substantially effect themeasurements of biological probes, such as microcantilevers which arevery sensitive to flow conditions.

The flow inside squeezed thin films, such as the flow inside isothermaloscillatory squeezed films with fluid density varying according to thepressure, has been studied. See Langlois (1962) Quarterly of AppliedMath. XX:131-150, which is herein incorporated by reference. The heattransfer inside squeezed thin films (not oscillatory type) has beenanalyzed. See Hamza (1992) J. Phys. D: Appl. Phys. 25:1425-1431,Bhattacharyya et al. (1996) Numerical Heat Transfer, Part A 30:519-532,and Debbaut (2001) J. Non-Newtonian Fluid Mech. 98:15-31, which areherein incorporated by reference. The flow and heat transfer insideincompressible oscillatory squeezed thin films has been analyzed. SeeKhaled & Vafai (2002) Numerical Heat Transfer Part A 41:451-467, whichis herein incorporated by reference. The effects of internal pressurepulsations have been studied on flow and heat transfer inside channels.See Hemida et al. (2002) Int. J. Heat Mass Transfer 45:1767-1780, andJoshi et al. (1985) J. Fluid Mech. 156:291-300, which are hereinincorporated by reference.

Unfortunately, the prior art fails to account for the effects of bothinternal and external pressure pulsations on flow and heat transferinside thin films, wherein the gap thickness will be a function of bothpulsations.

Therefore, as provided herein, the upper substrate of a thin film wasconsidered to be subjected to both external squeezing effects and theinternal pressure pulsations. The influence of internal pressurepulsations on the displacement of the upper substrate was determined bythe theory of linear elasticity applied to the seal supporting thesubstrates of an incompressible non-isothermal thin film. The laminargoverning equations for flow and heat transfer were properlynon-dimensionalized and reduced into simpler equations. The resultingequations were then solved numerically to determine the effects ofexternal squeezing, internal pressure pulsations and the strength of theseal on the turbulence inside the disturbed thin films as well as onthermal characteristics of these thin films.

2A. Problem Formulation

A two dimensional thin film that has a small thickness, h, compared toits length, B, was considered. The x-axis was taken in the direction ofthe length of the thin film while y-axis was taken along the thicknessas shown in FIG. 10. The width of the thin film, D, was assumed to belarge enough such that two dimensional flow inside the thin film can beassumed. The lower substrate of the thin film was fixed (immobile andinflexible substrate) while the vertical motion of the upper substrate(mobile and inflexible substrate) was assumed to have sinusoidalbehavior when the thin film gap was not charged with the working fluid.This motion due to only external disturbances is expressed according tothe following Equation 15:h=h _(o)(1−β cos(γωt))  Eq. 15where

γ is the dimensionless frequency

β is the dimensionless upper substrate motion amplitude

ω is a reference frequency.

The fluid was assumed to be Newtonian with constant properties.

The general two-dimensional continuity, momentum and energy equationsfor the laminar thin film are given as follows:

$\begin{matrix}{{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}} = 0} & {{Eq}.\mspace{14mu} 16} \\{{\rho\left( {\frac{\partial u}{\partial t} + {u\frac{\partial u}{\partial x}} + {v\frac{u}{y}}} \right)} = {{- \frac{\partial p}{\partial x}} + {\mu\left( {\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}}} \right)}}} & {{Eq}.\mspace{14mu} 17} \\{{\rho\left( {\frac{\partial v}{\partial t} + {u\frac{\partial v}{\partial x}} + {v\frac{\partial v}{\partial y}}} \right)} = {{- \frac{\partial p}{\partial y}} + {\mu\left( {\frac{\partial^{2}v}{\partial x^{2}} + \frac{\partial^{2}v}{\partial y^{2}}} \right)}}} & {{Eq}.\mspace{14mu} 18} \\{{\rho\;{c_{p}\left( {\frac{\partial T}{\partial t} + {u\frac{\partial T}{\partial x}} + {v\frac{\partial T}{\partial y}}} \right)}} = {k\left( {\frac{\partial^{2}T}{\partial x^{2}} + \frac{\partial^{2}T}{\partial y^{2}}} \right)}} & {{Eq}.\mspace{14mu} 19}\end{matrix}$where

T is the fluid temperature

ρ is the density

p is the pressure

μ is the dynamic viscosity

c_(p) is the specific heat

k is the thermal conductivity of the fluid

Equations 16-19 are non-dimensionalized using the followingdimensionless variables:

$\begin{matrix}{X = \frac{x}{B}} & {{{Eq}.\mspace{11mu} 20}a} \\{Y = \frac{y}{h_{o}}} & {{{Eq}.\mspace{14mu} 20}b} \\{\tau = {\omega\; t}} & {{{Eq}.\mspace{14mu} 20}c} \\{U = \frac{u}{\left( {{\omega\; B} + V_{o}} \right)}} & {{{Eq}.\mspace{14mu} 20}d} \\{V = \frac{v}{h_{o}\omega}} & {{{Eq}.\mspace{14mu} 20}e} \\{\Pi = \frac{p - p_{e}}{{\mu\left( {\omega + \frac{V_{o}}{B}} \right)}ɛ^{- 2}}} & {{{Eq}.\mspace{14mu} 20}f}\end{matrix}$where

T₁ is the inlet temperature of the fluid

V_(o) is a constant representing a reference dimensional velocity

As provided in the above equations, ΔT is equal to T₂−T₁ for constantwall temperature conditions (CWT), T₂ will be the temperature of bothlower and upper substrates, and is equal to

$\frac{q\; h_{o}}{k}$for uniform wall heat flux conditions (UHF). The variables X, Y, τ, U,V, Π and θ are the dimensionless forms of x, y, t, u, v, p and Tvariables, respectively. The above transformations except fordimensionless temperature have been used in the art along with theperturbation parameter ε,

$ɛ = {\frac{h_{o}}{B}.}$See Langlois (1962) Quarterly of Applied Math. XX: 131-150, which isherein incorporated by reference.

Most flows inside thin films are laminar and could be creep flowsespecially in lubrications and biological applications. Therefore, thelow Reynolds numbers flow model was adopted here. The application ofthis model to Equations 16-19 results in the following reducednon-dimensionalized equations:

$\begin{matrix}{U = {\frac{1}{2}\frac{\partial\Pi}{\partial X}(Y)\left( {Y - H} \right)}} & {{Eq}.\mspace{14mu} 21} \\{{\frac{\partial U}{\partial X} + {\frac{\sigma}{12}\frac{\partial V}{\partial Y}}} = 0} & {{Eq}.\mspace{14mu} 22} \\{{\frac{\partial}{\partial X}\left( {H^{3}\frac{\partial\Pi}{\partial X}} \right)} = {\sigma\frac{\partial H}{\partial\tau}}} & {{Eq}.\mspace{14mu} 23} \\{{P_{S}\left( {\frac{\partial\theta}{\partial\tau} + {\frac{12}{\sigma}U\frac{\partial\theta}{\partial X}} + {V\frac{\partial\theta}{\partial y}}} \right)} = {{ɛ^{2}\frac{\partial^{2}\theta}{\partial X^{2}}} + \frac{\partial^{2}\theta}{\partial y^{2}}}} & {{Eq}.\mspace{14mu} 24}\end{matrix}$where

σ is the squeezing number

P_(S) is the thermal squeezing parameter

The squeezing number and the thermal squeezing parameter are defined as:

$\begin{matrix}{\sigma = \frac{12}{1 + \frac{V_{o}}{\omega\; B}}} & {{{Eq}.\mspace{14mu} 25}a} \\{P_{S} = \frac{\rho\; c_{p}h_{o}^{2}\omega}{k}} & {{{Eq}.\mspace{14mu} 25}b}\end{matrix}$

The inlet dimensionless pulsating pressure is considered to have thefollowing relation:Π_(i)=Π_(o)(1+β_(p) sin(γ_(p) ωt+φ _(p)))  Eq. 26where

β_(p) is the dimensionless amplitude in the pressure

Π_(i) is the inlet dimensionless pressure

Π_(o) is the mean dimensionless pressure

γ_(p) is the dimensionless frequency of the pressure pulsationsparameter

φ_(p) is a phase shift angle parameter

Due to both pulsations in internal pressure and external disturbances,the dimensionless film thickness H, (H=h/h_(o)), can be represented byEquation 27 by noting the principle of superposition:H=1−β cos(γωt)+H _(p)  Eq. 27where H_(p) is the dimensionless deformation of the seals resulting frompulsations in the internal pressure.

The lower substrate was assumed to be fixed (immobile and inflexiblesubstrate) and that the upper substrate (mobile and inflexiblesubstrate) of the thin film is rigid such that the magnitude of thedeformation in the seals is similar to displacement of the uppersubstrate (mobile and inflexible substrate). The dimensionlessdeformation in the seals due to variations in the external pressure isthe second term of Equation 27 on the right. The dimensionless frequencyγ is allowed to be different than γ_(p).

The dimensionless pressure gradient inside the thin film as a result ofthe solution to the Reynolds Equation 23 is:

$\begin{matrix}{\frac{\partial\Pi}{\partial X} = {{\frac{\sigma}{H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {X - \frac{1}{2}} \right)} - {\Pi_{o}\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$

The reference velocity V_(o) that was used to define the dimensionlesspressure, axial dimensionless velocity and the squeezing number wastaken to be related to the average velocity, u_(m), inside the thin filmat zero β and β_(p) and the dimensionless thickness of the thin filmthat results from the application of the corresponding inlet meanpressure, H_(m), through the following relation:

$\begin{matrix}{V_{o} = \frac{u_{m}}{H_{m}^{2}}} & {{Eq}.\mspace{14mu} 29}\end{matrix}$

The previous scaled reference velocity is only a function of the meanpressure, viscosity and the reference dimensions of the thin film andresults in the following relation between the inlet mean dimensionlesspressure to the squeezing number:Π_(o)=12−σ  Eq. 30

Accordingly, the dimensionless pressure gradient, the dimensionlesspressure and the average dimensionless pressure Π_(AVG) inside the thinfilm were related to the squeezing number through the followingequations:

$\begin{matrix}{\frac{\partial{\Pi\left( {X,\tau} \right)}}{\partial X} = {{\frac{\sigma}{H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {X - \frac{1}{2}} \right)} - {\left( {12 - \sigma} \right)\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 31} \\{{\Pi\left( {X,\tau} \right)} = {{\frac{\sigma}{2H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {X^{2} - X} \right)} - {\left( {12 - \sigma} \right)\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)\left( {X - 1} \right)}}} & {{Eq}.\mspace{14mu} 32} \\{{\Pi_{AVG}(\tau)} = {{{- \frac{\sigma}{12H^{3}}}\frac{\mathbb{d}H}{\mathbb{d}\tau}} + {\frac{\left( {12 - \sigma} \right)}{2}\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 33}\end{matrix}$

The displacement of the upper substrate due internal pressure pulsationswas related to the Π_(AVG) through the theory of linear elasticity bythe following relation:H _(p) =F _(n)Π_(AVG)  Eq. 34where F_(n) is equal to

$\begin{matrix}{F_{n} = \frac{\mu\left( {V_{o} + {\omega\; B}} \right)}{E\; ɛ^{2}d_{s}}} & {{Eq}.\mspace{14mu} 35}\end{matrix}$

The parameters E and d_(s) in the previous equation are the modulus ofelasticity of the flexible seals of the present invention and acharacteristic dimension for the seal, respectively. The quantity d_(s)is equal to the effective diameter of the seal's cross section times theratio of the length of the seals divided by the thin film width. Theeffective diameter for seals having square cross section is equal toh_(o). The term F_(n) will be called the fixation number of the thinfilm.

The fixation parameter F_(n) represents a ratio between viscous shearforce inside thin films to the elastic forces of the flexible seals.Moreover, Equation 34 is based on the assumption that transient behaviorof the seal's deformation is negligible. The values of F_(n) are about0.001 to about 0.1 for long thin films supported by flexible seals.

The first set of dimensionless boundary conditions used were forconstant wall temperatures (CWT) at both the lower and the uppersubstrates while the second set used assumed that the lower substratewas at uniform wall heat flux conditions (UHF) and the upper substrateis insulated. As such the dimensionless boundary conditions can bewritten as:

$\begin{matrix}\begin{matrix}{C\; W\; T} & {{{\theta\left( {X,Y,0} \right)} = 0},{{\theta\left( {0,Y,\tau} \right)} = 0},{{\theta\left( {X,0,\tau} \right)} = 1}} \\\; & {{{\theta\left( {X,H,\tau} \right)} = 1},{{\frac{\partial}{\partial X}\left( \frac{1 - {\theta\left( {1,Y,\tau} \right)}}{1 - {\theta_{m}\left( {1,\tau} \right)}} \right)} = 0}}\end{matrix} & {{Eq}.\mspace{14mu} 36} \\\begin{matrix}{U\; H\; F} & {{{\theta\left( {X,Y,0} \right)} = 0},{{\theta\left( {0,Y,\tau} \right)} = 0},{\frac{\partial{\theta\left( {X,0,\tau} \right)}}{\partial Y} = {- 1}}} \\\; & {{\frac{\partial{\theta\left( {X,H,\tau} \right)}}{\partial Y} = 0},} \\\; & {\frac{\partial{\theta\left( {1,Y,\tau} \right)}}{\partial X} = {\frac{\sigma}{12U_{m}}\left( {\frac{1}{P_{S}H} - \frac{\partial{\theta\left( {1,Y,\tau} \right)}}{\partial\tau}} \right)}}\end{matrix} & {{Eq}.\mspace{14mu} 37}\end{matrix}$

The last condition of Equation 36 is based on the assumption that theflow at the exit of the thin film is thermally fully developed.Moreover, the last thermal condition of Equation 37 was derived based onan integral energy balance at the exit of the thin film realizing thatthe axial conduction is negligible at the exit. The calculated thermalparameters considered were the Nusselt numbers at the lower and uppersubstrates, and the dimensionless heat transfer from the upper and lowersubstrates, Θ, for CWT conditions, which are defined according to thefollowing equations:

$\begin{matrix}{\;\begin{matrix}{C\; W\; T} & {{{{Nu}_{U}\left( {X,\tau} \right)} \equiv \frac{h_{c}h_{o}}{k}} = {\frac{1}{1 - {\theta_{m}\left( {X,\tau} \right)}}\frac{\partial{\theta\left( {X,H,\tau} \right)}}{\partial Y}}} \\\; & {{{{Nu}_{L}\left( {X,\tau} \right)} \equiv \frac{h_{c}h_{o}}{k}} = {\frac{- 1}{1 - {\theta_{m}\left( {X,\tau} \right)}}\frac{\partial{\theta\left( {X,0,\tau} \right)}}{\partial Y}}}\end{matrix}} & {{Eq}.\mspace{14mu} 38} \\\begin{matrix}{U\; H\; F} & {{\Theta\left( {X,\tau} \right)} = \left( {\frac{\partial{\theta\left( {X,H,\tau} \right)}}{\partial Y} - \frac{\partial{\theta\left( {X,0,\tau} \right)}}{\partial Y}} \right)} \\\; & {{{{Nu}_{1}\left( {X,\tau} \right)} \equiv \frac{h_{c}h_{o}}{k}} = \frac{1}{{\theta\left( {X,0,\tau} \right)} - {\theta_{m}\left( {X,\tau} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} 39}\end{matrix}$where θm and Um are the dimensionless mean bulk temperature and thedimensionless average velocity at a given section and are defined asfollows:

$\begin{matrix}{{{\theta_{m}\left( {X,\tau} \right)} = {\frac{1}{{U_{m}\left( {X,\tau} \right)}H_{0}}{\int_{0}^{H}{{U\left( {X,Y,\tau} \right)}{\theta\left( {X,Y,\tau} \right)}{\mathbb{d}Y}}}}}\mspace{79mu}{{U_{m}\left( {X,\tau} \right)} = {\frac{1}{H}{\int_{0}^{H}{{U\left( {X,Y,\tau} \right)}{\mathbb{d}Y}}}}}} & {{{{Eqs}.\mspace{14mu} 40}a},{40b}}\end{matrix}$

Due to symmetric flow and thermal conditions for CWT, Nusselt numbers atlower and upper substrates were expected to be equal.

2B. Numerical Methods

The dimensionless thickness of the thin film was determined by solvingEquations 27, 33 and 34 simultaneously. Accordingly, the velocity field,U and V, was determined from Equations 21 and 22. The reduced energyequation, Equation 24, was then solved using the Alternative DirectionImplicit techniques (ADI) known in the art by transferring the problemto one with constant boundaries using the following transformations:

${\tau^{*} = \tau},{\xi = {{X\mspace{14mu}{and}\mspace{14mu}\eta} = {\frac{Y}{H}.}}}$Iterative solution was employed for the ξ-sweep of the energy equationfor CWT conditions so that both the energy equation and the exit thermalcondition, last condition of Equation 36, are satisfied. The values of0.008, 0.03, 0.002 were chosen for Δξ, Δη and Δτ*.2C. Effects of Pressure Pulsations on the Dimensionless Film Thickness

FIG. 11 and FIG. 12 describe the importance of the fixation number F_(n)on the dimensionless film thickness H and the dimensionless normalvelocity at the upper substrate V(X,H,τ), respectively. As F_(n)increases, H and absolute values of V(X,H,τ) increase. Soft (flexibleseals) fixations have large F_(n) values. Increases in the viscosity andflow velocities or a decrease in the thin film thickness, perturbationparameter and the seal's modulus of elasticity increase the value ofF_(n) as provided by Equation 35.

The effects of pressure pulsations on H are clearly seen for largevalues of F_(n) as shown in FIG. 11 and FIG. 12. At these values, thefrequency of the local maximum or minimum of H is similar to thefrequency of the pressure pulsations as seen from FIG. 11. Further, thedegree of turbulence at the upper substrate is increased when F_(n)increases as shown in FIG. 12. The degree of turbulence at the uppersubstrate refers to the degree of fluctuations at the upper substrateand the number of local maximum and minimum in V(X,H,τ). This is alsoobvious when the values of γ_(p) increase as shown in FIG. 13. Theincrease in turbulence level at the upper substrate may produce backflows inside the thin film at large values of γ_(p), which will have aneffect on the function of a thin film such as those used as chambers indetection and sensing devices.

For σ=12 where the time average of the average gage pressure inside thethin film is zero, the variation in H decreases as F_(n) increases. Thiseffect can be seen from Equation 33 and Equation 34 and will causereductions in the flow and in the cooling process. However, the meanvalue of Π_(AVG) is always greater than zero for other values of σ whichcauses an increase in the mean value of H as F_(n) increases resultingin an increase in the mean value of the flow rate inside the thin film.

FIG. 14 shows the effects of the squeezing number σ on H. Small valuesof σ indicates that the thin film is having relatively large inlet flowvelocities and therefore large pressure gradients and large values ofΠ_(o). Accordingly, H increases as σ decreases as seen in FIG. 14.Further, the degree of turbulence at the upper substrate increases as σdecreases. This is shown in FIG. 15. The changes in the pressure phaseshift results in similar changes in the dimensionless thin filmthickness phase shift as shown in FIG. 16.

2D. Effects of Pressure Pulsations on Heat Transfer Characteristics ofthe Thin Film

FIG. 17 and FIG. 18 illustrate the effects of F_(n) and P_(S) on thedimensionless mean bulk temperature θ_(m) and the average lowersubstrate temperature θ_(W), average of θ(X,0,τ), for constant walltemperature CWT and uniform heat flux UHF conditions, respectively. AsF_(n) increases when softer flexible seals are used, the inducedpressure forces inside the thin film due to internal pressure pulsationswill increase the displacement of the upper substrate (mobile andinflexible substrate) as shown before. This enables the thin film toreceive larger flow rates since the insulating assemblies in thesefigures have similar values for the dimensionless pressure at the inlet.Thus, there is more cooling to the substrates results as F_(n) increasesresulting in a decrease in the θ_(m) and average θ_(W) values and theircorresponding fluctuations for CWT and UHF conditions, respectively. Theeffect of the thermal squeezing parameter P_(S) on the cooling processis also shown in FIG. 17 and FIG. 18. The cooling at the substrates isenhanced as P_(S) increases.

FIG. 19 and FIG. 20 show the effects of F_(n) on the Nusselt number atthe lower substrate Nu_(L) for CWT and UHF conditions, respectively. Theirregularity in Nu_(L) decrease as F_(n) decreases because the uppersubstrate will not be affected by the turbulence in the flow if theflexible seals have relatively large modulus of elasticity. In otherwords, the induced flow due to the upper substrate motion is reduced asF_(n) decreases resulting in less disturbances to the flow inside thethin film. This can be seen in FIG. 21 for UHF conditions where Nu_(L)reaches a constant value at low values of F_(n) after a certain distancefrom the inlet. The values of Nu_(L) and the corresponding fluctuationsare noticed to decrease as F_(n) increases.

FIG. 22 and FIG. 23 illustrate the effects of dimensionless frequency ofthe inlet pressure pulsations γ_(p) on the average dimensionless heattransferred from the substrates Θ and the average θ_(W) for CWT and UHFconditions, respectively. The figures show that the mean value of Θ andθ_(W) are unaffected by γ_(p) and that the frequency of the averagevalues of Θ and θ_(W) increase as γ_(p) increases. FIG. 24 describes theeffects of γ_(p) on the fluctuation in the average Θ and θ_(W), half thedifference between the maximum and the minimum values of the average Θand θ_(W). The effects of γ_(p) on the fluctuation in the average Θ, δΘ,and the fluctuation in the average θ_(W), δθ_(W), are more pronounced atlower values of γ_(p) as shown in FIG. 24.

Flow and heat transfer inside externally oscillatory squeezed thin filmssupported by flexible seals in the presence of inlet internal pressurepulsations were analyzed. The governing laminar continuity, momentum andenergy equations were properly non-dimensionalized and reduced tosimpler forms for small Reynolds numbers. The reduced equations weresolved by the alternative direction implicit (ADI) method. Theturbulence level at the upper substrate increases by increases in boththe fixation number and the frequency of the internal pressurepulsations. However, an increase in the squeezing number decreases theturbulence level at the upper substrate. The fluid temperatures and thecorresponding fluctuations were found to decrease when the fixationnumber and the thermal squeezing parameter were increased for both CWTand UHF conditions. Finally, fluctuations in the heat transfer and thefluid temperatures were more pronounced at lower frequency of internalpressure pulsations.

3. Control of Exit Flow and Thermal Conditions Using Two-Layered ThinFilms Supported by Flexible Complex Seals

Although thin films are characterized by having laminar flows withrelatively low Reynolds numbers leading to stable hydrodynamicperformance, the thickness of the thin films is small enough such thatsmall disturbances at one of the boundaries may cause a significantsqueezing effect at the boundary. See e.g. Langlois (1962) Quarterly ofApplied Math. XX:131-150 (flow inside isothermal oscillatory squeezedfilms with fluid density varying with the pressure), Khaled & Vafai(2002) Numerical Heat Transfer, Part A 41:451-467 and Khaled & Vafai(2003) Int. J. Heat and Mass Transfer 46:631-641 (flow and heat transferinside incompressible thin films having a prescribed oscillatorysqueezing at one of their boundaries), and Khaled & Vafai (2002) Int. J.Heat and Mass Transfer 45:5107-5115 (internal pressure through theelastic behavior of the supporting seal), which are herein incorporatedby reference.

Recently, the situation where the squeezing effect at the free substrateis initiated by thermal effects was studied. See Khaled & Vafai (2003)ASME J. Heat Transfer 125:916-925, which is herein incorporated byreference. As provided herein, flexible seals with closed cavities ofstagnant fluids having a relatively large volumetric thermal expansioncoefficient, flexible complex seal, were studied. Flexible complex sealsin a single layer thin film can cause flooding of the coolant when thethermal load of the thin film is increased over its projected capacity.As a result, an enhancement in the cooling process is attainedespecially if ultrafine suspensions are present in the coolant, a fluidthat exhibits high heat transfer performance. Ultrafine suspensions inthe fluid such as copper or aluminum particles with diameters of ordernanometer are found to enhance the effective thermal conductivity of thefluid. See Eastman et al. (2001) Applied Physics Letters 78: 718-720,which is herein incorporated by reference.

As provided herein, the flow and heat transfer inside an oscillatorydisturbed two-layered thin film channel supported by flexible complexseals in the presence of suspended ultrafine particles was studied.Oscillatory generic disturbances were imposed on the two-layered thinfilm channels supported by flexible complex seals in the presence ofsuspended ultrafine particles, which correspond to disturbances in theupper substrate temperature and in the inlet pressure of the secondaryfluid layer. The governing continuity, momentum and energy equations forboth layers were non-dimensionalized and categorized for small Reynoldsnumbers and negligible axial conduction. The deformation of thesupporting seals was linearly related to both the pressure differenceacross the two layers and the upper substrate's temperature based on thetheory of the linear elasticity and the principle of the volumetricthermal expansion of the stagnant fluid filling the closed cavities ofthe flexible complex seals.

As provided herein, the flow rate and heat transfer in the main thinfilm channel can be increased by an increase in the softness of theseals, the thermal squeezing parameter, the thermal dispersion effectand the total thickness of two-layered thin film. However, the flow rateand heat transfer in the main thin film channel decrease as thedimensionless thermal expansion coefficient of the seals and thesqueezing number of the primary fluid layer increase. Both the increasein thermal dispersion and the thermal squeezing parameter for thesecondary fluid layer were found to increase the stability of theintermediate or the mobile and inflexible substrate. Furthermore, thetwo-layered thin film channel was found to be more stable when thesecondary fluid flow was free of pulsations or it had relatively a largepulsating frequency. Finally, the proposed two-layered thin filmsupported by flexible complex seals, unlike other controlling systems,does not require additional mechanical control or external coolingdevices, i.e. is self-regulating for the flow rate and temperature of aprimary fluid layer.

The following Table 6 provides the various symbols and meanings used inthis section:

TABLE 6 B thin film length C_(F) correction factor for the volumetricthermal expansion coefficient c_(p) specific heat of the fluid D widthof the thin film E* softness index of seals supporting the intermediateor mobile and inflexible substrate G width of closed cavity containingstagnant fluid H_(t) dimensionless total thickness of the two-layeredthin film F_(T) dimensionless coefficient of the thermal expansion forthe complex seal H, h, h_(o) dimensionless, dimensional and referencethin film thickness h_(c) convective heat transfer coefficient K*effective stiffness of the sealing k thermal conductivity of the fluidk_(o) reference thermal conductivity of the fluid Nu lower substrate'sNusselt number P_(S) thermal squeezing parameter p fluid pressure q_(o)reference heat flux at the lower substrate for UHF T, T_(o) temperaturein fluid and the inlet temperature t time V_(o) reference axial velocityU, U_(m) dimensionless axial and average axial velocities u dimensionalaxial velocity V, v dimensionless and dimensional normal velocities X, xdimensionless and dimensional axial coordinates Y, y dimensionless anddimensional normal coordinates α thermal diffusivity of the fluid β_(q)dimensionless amplitude of the thermal load β_(p) dimensionlessamplitude of the pressure β_(T) coefficient of volumetric thermalexpansion ε perturbation parameter φ_(p) phase shift angle γdimensionless frequency of the thermal load γ_(p) dimensionlessfrequency of the internal pressure μ dynamic viscosity of the fluid θ,θ_(m) dimensionless temperature and dimensionless mean bulk temperatureρ density of the fluid τ, τ* dimensionless time σ squeezing number ωreciprocal of a reference time (reference squeezing frequency) ηvariable transformation for the dimensionless Y-coordinate λdimensionless dispersion parameter Π dimensionless pressure Π_(n)dimensionless inlet pressure Λ reference lateral to normal velocityratio Subscripts i i^(th) layer l lower substrate P due to pressure Tdue to thermal expansion u upper substrate

The present invention provides flexible complex seals. The flexiblecomplex seals may be used in two-layered thin films are utilized inorder to regulate the flow rate of the primary fluid layer such thatexcessive heating in the second layer results in a reduction in theprimary fluid flow rate. The flexible complex seals of the presentinvention may be used in internal combustion applications where the fuelflow rate should be reduced as an engine gets overheated. The flexiblecomplex seals of the present invention may be used to minimizebimaterial effects of many biosensors that are sensitive to heat andflow conditions. See Fritz et al. (2000) Science 288:316-318, which isherein incorporated by reference.

3A. Problem Formulation and Analysis

FIG. 25 shows a two-layered thin film supported by flexible complexseals. The lower layer contains the primary fluid flow passage where thelower substrate is fixed (immobile and inflexible substrate) and theupper substrate is insulated and free to move in the vertical direction(mobile and inflexible substrate). The primary fluid flow is that of afluid sample, such as the fuel flow or fuel-air mixture prior tocombustion or flow of a biofluid in a fluidic cell. The upper layer ofthe thin film contains a secondary fluid flow parallel or counter to theprimary fluid flow direction. This flow can have similar properties asthe primary fluid flow. This insulating assembly is suitable for fluidiccell applications since inlet pressure pulsations will be equal acrossthe intermediate substrate, thereby eliminating disturbances at theintermediate substrate. The secondary fluid flow, however, can havedifferent properties than the primary fluid flow. For example, when thesecondary fluid flow is initiated from external processes such as flowof combustion residuals or the engine coolant flow.

The heat flux of the upper substrate can be independent of the primaryfluid flow or can be the result of external processes utilizing theprimary fluid flow as in combustion processes. The latter can be usedfor controlling the primary fluid flow conditions while the former maymodel the increase in the ambient temperature in a fluidic cellapplication, thereby preventing an increase in the average fluidtemperature in an ordinary fluidic cell avoiding a malfunctioning of adevice such as a biosensor.

The sealing assembly of the upper layer contains flexible complex seals,closed cavities filled with a stagnant fluid having a relatively largevolumetric thermal expansion coefficient. The upper layer also containsflexible seals in order to allow the intermediate substrate to move inthe normal direction. Any excessive heating at the upper substrateresults in an increase in the upper substrate's temperature such thatthe stagnant fluid becomes warmer and expands. This expansion along withthe increase in inlet pressure in the upper layer, if present, cause theintermediate substrate to move downward. Thus, a compression in the filmthickness of the lower layer is attained resulting in reduction in massflow rate within the primary fluid flow compartment. This insulatingassembly may be used to control combustion rates since part of theexcessive heating and increased pressure due to deteriorated combustionconditions can be utilized to prescribe the heat flux at the uppersubstrate. Thus, the flow rate of the fuel in the primary fluid layercan be reduced and combustion is controlled.

In fluidic cells, excessive heating at the upper substrate causescompression to the primary fluid layer's thickness. Thus, averagevelocity in the primary fluid layer increases, when operated at constantflow rates, enhancing the convective heat transfer coefficient. Thiscauses the average fluid temperature to approach the lower substratetemperature, thereby reducing the bimaterial effects. When it isoperated at a constant pressure or at a constant velocity, thecompression of the primary fluid layer due to excessive heating at theupper substrate reduces the flow rate. Thus, the fluid temperaturesapproach the lower substrate temperature at a shorter distance. As such,bimaterial effects are also reduced. The flexible seals can be placedbetween guiders as shown in FIG. 25B. The use of guiders for theflexible seals, including flexible complex seals, of the presentinvention minimize side expansion and maximize the transverse thin filmthickness expansion.

As provided herein, upper and lower thin films that have smallthicknesses h₁ and h₂, respectively, compared to their length B andtheir width D₁ and D₂, respectively, were analyzed. The x-axis for eachlayer is taken along the axial direction of the thin film while y-axisfor each layer is taken along its thickness as shown in FIG. 25B.Further, the film thickness was assumed to be independent of the axialdirection. For example, as in symmetric thin films having a fluidinjected from the center as shown in FIG. 25A.

Both lower and upper substrates were assumed to be fixed (immobile andinflexible substrates) while the intermediate substrate was free to moveonly in the normal direction due to the use of flexible complex seals(mobile and inflexible substrate). The generic motion of theintermediate substrate due to both variations of the stagnant fluidtemperature in the secondary fluid flow passage and the induced internalpressure pulsations within both primary fluid and secondary fluid flowpassages is expressed according to the following Equation 41:

$\begin{matrix}{H_{1} = {\frac{h_{1}}{h_{o}} = \left( {1 + H_{T} + H_{p}} \right)}} & {{Eq}.\mspace{14mu} 41}\end{matrix}$where

h_(o) is a reference thickness for the primary fluid passage

H₁ is the dimensionless motion of the intermediate substrate

H_(T) is the dimensionless motion of the intermediate substrate due tothe volumetric thermal expansion of the stagnant fluid

H_(p) is the dimensionless motion of the intermediate substrate due tothe deformation in seals as a result of the internal pressure.

The fluid was assumed to be Newtonian having constant average propertiesexcept for the thermal conductivity. The general two-dimensionalcontinuity, momentum and energy equations for a laminar thin film aregiven as follows:

$\begin{matrix}{\mspace{79mu}{{\frac{\partial u_{i}}{\partial x_{i}} + \frac{\partial v_{i}}{\partial y_{i}}} = 0}} & {{Eq}.\mspace{14mu} 42} \\{\mspace{79mu}{{\rho_{i}\left( {\frac{\partial u_{i}}{\partial t} + {u_{i}\frac{\partial u_{i}}{\partial x_{i}}} + {v_{i}\frac{\partial u_{i}}{\partial y_{i}}}} \right)} = {{- \frac{\partial p_{i}}{\partial x_{i}}} + {\mu_{i}\left( {\frac{\partial^{2}u_{i}}{\partial x_{i}^{2}} + \frac{\partial^{2}u_{i}}{\partial y_{i}^{2}}} \right)}}}} & {{Eq}.\mspace{14mu} 43} \\{\mspace{79mu}{{\rho_{i}\left( {\frac{\partial v_{i}}{\partial t} + {u_{i}\frac{\partial v_{i}}{\partial x_{i}}} + {v_{i}\frac{\partial v_{i}}{\partial y_{i}}}} \right)} = {{- \frac{\partial p_{i}}{\partial y_{i}}} + {\mu_{i}\left( {\frac{\partial^{2}v_{i}}{\partial x_{i}^{2}} + \frac{\partial^{2}v_{i}}{\partial y_{i}^{2}}} \right)}}}} & {{Eq}.\mspace{14mu} 44} \\{{\left( {\rho\; c_{p}} \right)_{i}\left( {\frac{\partial T_{i}}{\partial t} + {u_{i}\frac{\partial T_{i}}{\partial x_{i}}} + {v_{i}\frac{\partial T_{i}}{\partial y_{i}}}} \right)} = {{\frac{\partial}{\partial x_{i}}\left( {k_{i}\frac{\partial T_{i}}{\partial x_{i}}} \right)} + {\frac{\partial}{\partial y_{i}}\left( {k_{i}\frac{\partial T_{i}}{\partial y_{i}}} \right)}}} & {{Eq}.\mspace{14mu} 45}\end{matrix}$where

T is the fluid temperature

u is the dimensional axial velocity

v is the dimensional normal velocity

ρ is the average fluid density

p is pressure

μ is the average fluid dynamic viscosity

cp is the average specific heat of the fluid

k is the thermal conductivity of the fluid

When the fluid contains suspended ultrafine particles, these propertieswill be for the resulting dilute mixture so long as the diameter of theparticles is very small compared to h_(o). The index “i” is “1” whenanalyzing the primary fluid layer while it is “2” when analyzing thesecondary fluid layer. Equations 42-45 are non-dimensionalized using thefollowing dimensionless variables:

$\begin{matrix}{X_{i} = \frac{x_{i}}{B}} & {{{Eq}.\mspace{14mu} 46}a} \\{Y_{i} = \frac{y_{i}}{h_{o}}} & {{{Eq}.\mspace{14mu} 46}b} \\{\tau = {\omega\; t}} & {{{Eq}.\mspace{14mu} 46}c} \\{U_{i} = \frac{u_{i}}{\left( {{\omega\; B} + V_{oi}} \right)}} & {{{Eq}.\mspace{14mu} 46}d} \\{V_{i} = \frac{v_{i}}{h_{o}\omega}} & {{{Eq}.\mspace{14mu} 46}e} \\{\Pi_{i} = \frac{p_{i} - p_{e\; i}}{{\mu_{i}\left( {\omega + \frac{V_{o\; i}}{B}} \right)}ɛ^{- 2}}} & {{{Eq}.\mspace{14mu} 46}f} \\{\theta_{1} = \frac{T_{1} - T_{1o}}{\left( {T_{w} - T_{1o}} \right)}} & {{{Eq}.\mspace{14mu} 46}g} \\{\theta_{2} = \frac{T_{2} - T_{2o}}{q_{o}{h_{o}/k_{2o}}}} & {{{Eq}.\mspace{14mu} 46}h}\end{matrix}$where

ω is the reference frequency of the disturbance

T_(1o) is the inlet temperature for the primary fluid flow

T_(2o) is the inlet temperature for the secondary fluid flow

T_(w) is the lower substrate temperature

p_(e) is the reference pressure which represents the exit pressure forboth layers

q_(o) is the reference heat flux at the upper substrate

k_(2o) is the stagnant thermal conductivity of the secondary fluid

V_(o1) is the reference dimensional velocity for the lower layer

V_(o2) is the reference dimensional velocity for the upper layer

ε is the perturbation parameter,

$ɛ = \frac{h_{o}}{B}$

The prescribed heat at the upper substrate, q_(u), as well as thedimensionless inlet pressure, Π_(2n), for the secondary fluid flow varyaccording to the following generic relationships:q _(u) =q _(o)(1+β_(q) sin(γωt))  Eq. 47Π_(2n)=Π_(2o)(1+β_(p) sin(γ_(p) ωt+φ _(p))) Eq. 48where

β_(q) is the dimensionless amplitude of upper substrate's heat flux

β_(p) is the dimensionless amplitude for the inlet pressure for thesecondary fluid flow

γ is the dimensionless frequency for the upper substrate heat flux

γ_(p) is the dimensionless frequency for the inlet pressure for thesecondary fluid layer

The variables X_(i), Y_(i), τ, U_(i), V_(i), Π_(i) and θ_(i) are thedimensionless forms of x_(i), y_(i), t, u_(i), v_(i), p_(i) and T_(i)variables, respectively.

For the two-layered thin film shown in FIG. 25A, the displacement of theintermediate substrate due to internal pressure variations was relatedto the difference in the average dimensionless pressure across theintermediate substrate through the theory of the linear elasticity by:

$\begin{matrix}{H_{p} = {{E_{1}^{*}\frac{\left( \Pi_{AVG} \right)_{1}}{\sigma_{1}}} - {E_{2}^{*}\frac{\left( \Pi_{AVG} \right)_{2}}{\sigma_{2}}}}} & {{Eq}.\mspace{14mu} 49}\end{matrix}$where (Π_(AVG))₁ and (Π_(AVG))₂ are the average dimensionless pressurein the primary fluid and the secondary fluid layers, respectively. Theparameter E_(i)* will be referred to as the softness index of thesupporting seal in layers “1” or “2” and will be denoted as E* whenE₁*=E₂*. It has the following functional form:

$\begin{matrix}{E_{i}^{*} = \frac{12L^{*}\mu_{i}\omega\; D_{i}}{K^{*}ɛ^{3}}} & {{Eq}.\mspace{14mu} 50}\end{matrix}$where K* is the effective stiffness of the seals that support theintermediate substrate. The dimensionless parameter L* is introduced toaccount for the elastic contribution of the intermediate substrate inthe calculation of the displacement.

As provided herein, the analysis was performed for relatively smallthermal load frequencies in order to ascertain that squeezing generatedflows are in the laminar regime. For these frequencies, Equation 49 wasapplicable and the inertia effect of the intermediate substrate wasnegligible. Moreover, the increase in the thickness due to a pressureincrease in the thin film causes a reduction in the stagnant fluidpressure. This action stiffens the insulating assembly. Therefore, thestiffness K* was considered to be the effective stiffness for theinsulating assembly and not for the seal itself. From the practicalpoint of view, the closed cavity width G was taken to be large enoughsuch that a small increase in the stagnant fluid pressure due to thethermal expansion can support the associated increase in the elasticforce on the seal.

The dimensionless displacement of the intermediate substrate due to thethermal expansion was related to the dimensionless average temperatureof the upper substrate, (θ_(u))_(AVG), by the following linearizedmodel:H _(T) =−F _(T)(θ_(u))_(AVG)  Eq. 51where F_(T) is named the dimensionless thermal expansion parameter andis equal to:

$\begin{matrix}{F_{T} = {A^{*}\frac{\beta_{T}q_{o}h_{o}}{k_{2o}}C_{F}}} & {{Eq}.\mspace{14mu} 52}\end{matrix}$

The coefficient A* depends on the closed cavities dimensions and theirgeometry. The parameter β_(T) is the volumetric thermal expansioncoefficient of the stagnant fluid in its approximate form:

${\beta_{T} \approx {\frac{1}{V_{so}}\frac{\left( {V_{s} - V_{s\; 1}} \right)}{\left( {T_{s} - T_{2o}} \right)}}}❘_{p_{s\; 1}}$evaluated at the pressure p_(s1) corresponding to the stagnant fluidpressure in the closed cavities when the secondary fluid flowtemperature was kept at inlet temperature of the secondary fluid layerT_(2o). The closed cavity volumes V_(so), V_(s1) and V_(s) represent theclosed cavity volume at the reference condition (h₂=h_(o)), the closedcavity volume when the pressure in the closed cavities is p_(s1) and theclosed cavity volume at normal operating conditions where the averagestagnant fluid temperature is T_(s), respectively. The factor C_(F)represents the volumetric thermal expansion correction factor. Thisfactor was introduced in order to account for the increase in thestagnant pressure due to the increase in the elastic force in the sealduring the expansion which tends to decrease the effective volumetricthermal expansion coefficient. It approaches one as the closed cavitywidth G increases and it can be determined theoretically using methodsknown in the art.

The parameter F_(T) is enhanced at elevated temperatures for liquids andat lower temperatures for gases because β_(T) increases for liquids anddecreases for gases as the temperature increases. Dimensionless thermalexpansion parameter is further enhanced by a decrease in k_(o), anincrease in q_(o), an increase in E_(i)* or an increase in h_(o).Equation 51 is based on the assumption that the stagnant fluidtemperature is similar to the average upper substrate temperature sinceclosed cavity surfaces were considered insulated except for the regionfacing the upper substrate in order to provide a maximum volumetricthermal expansion to the closed cavities. Moreover, the heat flux on theupper substrate was assumed to be applied to the portion that faces thesecondary fluid flow.

The thermal conductivity of the fluid was considered to vary with theflow speed in order to account for thermal dispersion effects whensuspended ultrafine particles were present in the secondary fluid flow.Induced squeezing effects at the intermediate substrate due to timevariations in the thermal load or inlet pulsative pressures wereexpected to enhance the heat transfer inside fluid layers due to thermaldispersion effects. To account for this increase, a linear model betweenthe effective thermal conductivity and the fluid speed was utilized asprovided by Equation 53. See Xuan & Roetzel (2000) Int. J. Heat and MassTransfer 43:3701-3707, which is herein incorporated by reference.k _(i)(X _(i) , Y _(i), τ)=(k _(o))_(i)(1+λ_(i)√{square root over (U ²(X_(i) , Y _(i), τ)+Λ_(i) ² V ²(X _(i) , Y _(i), τ))}{square root over (U²(X _(i) , Y _(i), τ)+Λ_(i) ² V ²(X _(i) , Y _(i), τ))})=(k_(o))_(i)φ_(i)(X _(i) , Y _(i), τ)  Eq. 53where λ_(i) and Λ_(i) are the dimensionless thermal dispersioncoefficient and reference squeezing to lateral velocity ratio which are:

$\begin{matrix}{\lambda_{i} = {{C_{i}^{*}\left( {\rho\; c_{p}} \right)}_{f\; i}{h_{o}\left( {V_{o\; i} + {\omega\; B}} \right)}}} & {{{Eq}.\mspace{14mu} 54}a} \\{\Lambda_{i} = \frac{ɛ\;\sigma_{i}}{12}} & {{{Eq}.\mspace{14mu} 54}b}\end{matrix}$

The coefficient C* depends on the diameter of the ultrafine particle,its volume fraction and both fluid and the particle properties. Theparameter (ρc_(p))_(fi) is the density times the specific heat of thefluid resulting from the mixture of the pure fluid and the ultrafineparticles suspensions within the i^(th) layer while (k_(o))_(i) is thestagnant thermal conductivity of the working fluid in the i^(th) layerthat contains ultrafine particles. This stagnant thermal conductivity isusually greater than the thermal conductivity of the pure fluid. SeeEastman et al. (2001) Applied Physics Letters 78:718-720, which isherein incorporated by reference. All the fluid properties that appearin Equations 42-45 should be replaced by the effective mixtureproperties which are functions of the pure fluid and the particles andthat the diameter of the ultrafine particles are so small that theresulting mixture behaves as a continuum fluid. See Xuan & Roetzel(2000) Int. J. Heat and Mass Transfer 43:3701-3707, which is hereinincorporated by reference.

Flows inside thin films are in laminar regime and could be consideredcreep flows in certain applications as in lubrication and biologicalapplications. Therefore, the low Reynolds numbers flow model was adoptedand applied to Equations 42-44 and the results of dimensionalizing theenergy equation result in the following reduced non-dimensionalizedequations:

$\begin{matrix}{U_{i} = {\frac{1}{2}\frac{\partial\Pi_{i}}{\partial X}{H_{i}^{2}\left( \frac{Y_{i}}{H_{i}} \right)}\left( {\frac{Y_{i}}{H_{i}} - 1} \right)}} & {{Eq}.\mspace{14mu} 55} \\{V_{i} = {\frac{\mathbb{d}H_{i}}{\mathbb{d}\tau}\left( {{3\left( \frac{Y_{i}}{H_{i}} \right)^{2}} - {2\left( \frac{Y_{i}}{H_{i}} \right)^{3}}} \right)}} & {{Eq}.\mspace{14mu} 56} \\{\frac{\partial\Pi_{i}}{\partial Y_{i}} = 0} & {{Eq}.\mspace{14mu} 57} \\{{\frac{\partial}{\partial X_{i}}\left( {H_{i}^{3}\frac{\partial\Pi_{i}}{\partial X_{i}}} \right)} = {\sigma_{i}\frac{\partial H_{i}}{\partial\tau}}} & {{Eq}.\mspace{14mu} 58} \\{{\left( P_{s} \right)_{i}\left( {\frac{\partial\theta_{i}}{\partial\tau} + {\frac{12}{\sigma_{i}}U_{i}\frac{\partial\theta_{i}}{\partial X_{i}}} + {V_{i}\frac{\partial\theta_{i}}{\partial Y_{i}}}} \right)} = {\frac{\partial}{\partial Y_{i}}\left( {\phi_{i}\frac{\partial\theta_{i}}{\partial Y_{i}}} \right)}} & {{Eq}.\mspace{14mu} 59}\end{matrix}$

The axial diffusion term in the dimensionalized energy equation,Equation 59, is eliminated because it is of order ε². The parametersσ_(i) and (P_(S))_(i) are called the squeezing number and the thermalsqueezing parameter, respectively, and are defined as:

$\begin{matrix}{\sigma_{i} = \frac{12}{1 + \frac{V_{oi}}{\omega\; B}}} & {{{Eq}.\mspace{14mu} 60}a} \\{\left( P_{S} \right)_{i} = \frac{\left( {\rho\; c_{p}} \right)h_{o}^{2}\omega}{k_{i}}} & {{{Eq}.\mspace{14mu} 60}b}\end{matrix}$

The dimensionless thickness of the lower layer and the upper layer aredefined as:

$\begin{matrix}{H_{1} = \frac{h_{1}}{h_{o}}} & {{{Eq}.\mspace{14mu} 61}a} \\{H_{2} = \frac{h_{2}}{h_{o}}} & {{{Eq}.\mspace{14mu} 61}b}\end{matrix}$

The reference thickness h_(o) can be determined using the force balanceacross the intermediate substrate due to the flow exit pressures of bothlayers at static conditions using methods known in the art. Thereference thickness h_(o) can be controlled by either varying flow exitpressures for each layer prior injecting of both flows, by a properselection to the undistorted thickness of the supporting seals in eachlayer or by using both, according to methods known in the art.Therefore, the dimensionless thicknesses H₁ and H₂ are related to eachother through the following relation as both lower and upper substratesare fixed (immobile and inflexible substrates):H ₁ +H ₂ =H _(t)  Eq. 62where H_(t) is a constant representing the dimensionless total thicknessof the two-layered thin film.

Two conditions will be imposed for the inlet flow rate of the primaryfluid layer. In applications that require minimizations of thermaleffects due to an increase in heat transfer from the environment such asfor fluidic cells of biological and chemical sensing devices, the inletflow rate for the lower layer is assumed to be constant and referred toas the CIF condition. However, constant inlet pressure was assumed tomodel flow of fluids in combustion applications such as flow of fuelprior to the mixing section and is referred as the CIP condition. Thepreviously defined reference velocities V_(o1) and V_(o2) represent thevelocity in the flow passages at zero values of the parameters E₁*, E₂*and F_(T). Accordingly, the inlet dimensionless pressures vary with thesqueezing numbers according to following relations for the CIPcondition:Π_(1n)=12−σ₁  Eq. 63Π_(2n)=(12−σ₂)(1+β_(p) sin(γ_(p)τ+φ_(p)))  Eq. 64

Therefore, the solution of the Reynolds equations for the CIP conditionwill reveal the following relationships for the dimensionless pressuregradient, the dimensionless pressure and the average dimensionlesspressure Π_(AVG) inside lower and upper layers:

$\begin{matrix}{\mspace{79mu}{\frac{\partial{\Pi_{1}\left( {X_{1},\tau} \right)}}{\partial X_{1}} = {{\frac{\sigma_{1}}{H_{1}^{3}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}\left( {X_{1} - \frac{1}{2}} \right)} - \left( {12 - \sigma_{1}} \right)}}} & {{{Eq}.\mspace{14mu} 65}a} \\{\frac{\partial{\Pi_{2}\left( {X_{2},\tau} \right)}}{\partial X_{2}} = {{\frac{\sigma_{2}}{H_{2}^{3}}\frac{\mathbb{d}H_{2}}{\mathbb{d}\tau}\left( {X_{2} - \frac{1}{2}} \right)} - {\left( {12 - \sigma_{2}} \right)\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 66} \\{\mspace{79mu}{{\Pi_{1}\left( {X_{1},\tau} \right)} = {{\frac{\sigma_{1}}{2H_{1}^{3}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}\left( {X_{1}^{2} - X_{1}} \right)} - {\left( {12 - \sigma_{1}} \right)\left( {X_{1} - 1} \right)}}}} & {{{Eq}.\mspace{14mu} 67}a} \\{{\Pi_{2}\left( {X_{2},\tau} \right)} = {{\frac{\sigma_{2}}{2H_{2}^{3}}\frac{\mathbb{d}H_{2}}{\mathbb{d}\tau}\left( {X_{2}^{2} - X_{2}} \right)} - {\left( {12 - \sigma_{2}} \right)\left( {X_{2} - 1} \right)\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 68} \\{\mspace{79mu}{\left( {\Pi_{AVG}(\tau)} \right)_{1} = {{{- \frac{\sigma_{1}}{12H_{1}^{3}}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}} + \frac{\left( {12 - \sigma_{1}} \right)}{2}}}} & {{{Eq}.\mspace{14mu} 69}a} \\{\left( {\Pi_{AVG}(\tau)} \right)_{2} = {{{- \frac{\sigma_{2}}{12H_{2}^{3}}}\frac{\mathbb{d}H_{2}}{\mathbb{d}\tau}} + {\frac{\left( {12 - \sigma_{2}} \right)}{2}\left( {1 + {\beta_{p}{\sin\left( {{\gamma_{p}\tau} + \varphi_{p}} \right)}}} \right)}}} & {{Eq}.\mspace{14mu} 70}\end{matrix}$

For the CIF condition, the dimensionless pressure gradient, thedimensionless pressure and the average dimensionless pressure Π_(AVG)inside lower layer were changed to the following:

$\begin{matrix}{\frac{\partial{\Pi_{1}\left( {X_{1},\tau} \right)}}{\partial X_{1}} = {{\frac{\sigma_{1}}{H_{1}^{3}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}X_{1}} - \frac{\left( {12 - \sigma_{1}} \right)}{H_{1}^{3}}}} & {{{Eq}.\mspace{14mu} 65}b} \\{{\Pi_{1}\left( {X_{1},\tau} \right)} = {{\frac{\sigma_{1}}{2H_{1}^{3}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}\left( {X_{1}^{2} - 1} \right)} - {\frac{\left( {12 - \sigma_{1}} \right)}{H_{1}^{3}}\left( {X_{1} - 1} \right)}}} & {{{Eq}.\mspace{14mu} 67}b} \\{\left( {\Pi_{AVG}(\tau)} \right)_{1} = {{{- \frac{\sigma_{1}}{3H_{1}^{3}}}\frac{\mathbb{d}H_{1}}{\mathbb{d}\tau}} + \frac{\left( {12 - \sigma_{1}} \right)}{2H_{1}^{3}}}} & {{{Eq}.\mspace{14mu} 69}\; b}\end{matrix}$3B. Thermal Boundary Conditions

The dimensionless initial and thermal boundary conditions for thepreviously defined problem were taken as follows:

$\begin{matrix}{{{\theta_{1}\left( {X_{1},Y_{1},0} \right)} = 0},{{\theta_{1}\left( {0,Y_{1},\tau} \right)} = 0},{{\theta_{1}\left( {X_{1},0,\tau} \right)} = 1},{\frac{\partial{\theta_{1}\left( {X_{1},H_{1},\tau} \right)}}{\partial Y_{1}} = 0}} & {{Eq}.\mspace{14mu} 71} \\{{{\theta_{2}\left( {X_{2},Y_{2},0} \right)} = 0},{{\theta_{2}\left( {0,Y_{2},\tau} \right)} = 0},{\frac{\partial{\theta_{2}\left( {X_{2},0,\tau} \right)}}{\partial Y_{2}} = {- \left( {1 + {\beta_{q}{\sin\left( {\gamma\;\tau} \right)}}} \right)}},{\frac{\partial{\theta_{2}\left( {X_{2},H_{2},\tau} \right)}}{\partial Y_{2}} = 0}} & {{Eq}.\mspace{14mu} 72}\end{matrix}$

Based on physical conditions, the intermediate substrate was taken to beinsulated and the Nusselt number at the lower and the upper substratesare defined as:

$\begin{matrix}{{{{Nu}_{1}\left( {X_{1},\tau} \right)} \equiv \frac{h_{c\; l}h_{o}}{k_{1}}} = {{- \frac{1}{1 - \theta_{1m}}}\frac{\partial{\theta_{1}\left( {X_{1},0,\tau} \right)}}{\partial Y_{1}}}} & {{Eq}.\mspace{14mu} 73} \\\begin{matrix}{{{{Nu}_{u}\left( {X_{2},\tau} \right)} \equiv \frac{h_{c\; u}h_{o}}{k_{2}}} = \frac{1}{{\theta_{2}\left( {X_{2},0,\tau} \right)} - {\theta_{2m}\left( {X_{2},\tau} \right)}}} \\{= \frac{1}{{\theta_{u}\left( {X_{2},\tau} \right)} - {\theta_{2m}\left( {X_{2},\tau} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} 74}\end{matrix}$where h_(c1) and h_(cu) are the convective heat transfer coefficientsfor the lower and upper substrates, respectively.

The quantities θ_(im) and U_(im) are the sectional dimensionless meanbulk temperature and the dimensionless average velocity for the i^(th)layer and are given as:

$\begin{matrix}{{{\theta_{i\; m}\left( {X_{i},\tau} \right)} = {\frac{1}{{U_{i\; m}\left( {X_{i},\tau} \right)}H_{i}}{\int_{0}^{H_{i}}{{U_{i}\left( {X_{i},Y_{i},\tau} \right)}{\theta_{i}\left( {X_{i},Y_{i},\tau} \right)}{\mathbb{d}Y_{i}}}}}}\mspace{79mu}{{U_{i\; m}\left( {X_{i},\tau} \right)} = {\frac{1}{H_{i}}{\int_{0}^{H_{i}}{{U_{i}\left( {X_{i},Y_{i},\tau} \right)}{\mathbb{d}Y_{i}}}}}}} & {{Eq}.\mspace{14mu} 75}\end{matrix}$where U_(im) is the dimensionless average velocity at a given sectionfor the i^(th) layer. For the primary fluid passage, the dimensionlessheat flux at a given section is defined as follows:

$\begin{matrix}{{\Theta\left( {X_{1},\tau} \right)} = {- \frac{\partial{\theta_{1}\left( {X_{1},0,\tau} \right)}}{\partial Y_{1}}}} & {{Eq}.\mspace{14mu} 76}\end{matrix}$3C. Dimensionless Flow Rate Parameter for the Primary Fluid Layer

The obtained dimensionless film thickness for the primary fluid layer H₁can be used to determine the dimensionless flow rate of the fluid in theprimary fluid passage at the mid section for the CIP condition. Thelatter is an important parameter should be controlled and is referred toas Ψ_(X=0.5) where X=0.5 denotes the location at X₁=0.5. This parametercan be calculated from the following relation:

$\begin{matrix}{\Psi_{X = 0.5} = {\frac{Q_{X = 0.5}}{\left( {V_{o\; 1} + {\omega\; B}} \right)h_{o}} = {\frac{\left( {12 - \sigma_{1}} \right)}{12}H_{1}^{3}}}} & {{Eq}.\mspace{14mu} 77}\end{matrix}$where Q_(X=0.5) is the dimensional flow rate at X=0.5 in the main thinfilm.3D. Numerical Procedure

The procedure for the numerical solution is summarized as follows:

-   1. Initially, a value for H_(T) is assumed.-   2. The dimensionless thicknesses for the lower and upper layers H₁    and H₂ are determined by solving Equations 41, 49, 62, 69, and 70    simultaneously, using an explicit formulation. The velocity field,    U_(i) and V_(i), is then determined from Equations 55, 56, 65, and    66.-   3. Reduced energy equations, Equation 59, are solved by first    transferring them to a constant boundary domain using the following    transformations:

${\tau^{*} = \tau},{\xi_{I} = {{X_{i}\mspace{14mu}{and}\mspace{14mu}\eta_{1}} = {\frac{Y_{i}}{H_{i}}.}}}$Tri-diagonal algorithm was implemented along with a marching scheme. SeeBlottner (1970) AIAA J. 8:193-205, which is herein incorporated byreference. Backward differencing was chosen for the axial convective andtransient terms and central differencing was selected for thederivatives with respect to η_(i). The values of 0.008, 0.03, 0.001 werechosen for Δξ_(i), Δη_(i) and Δτ*, respectively.

-   4. H_(T) is updated from Equation 51 and steps (2) to (4) is    repeated until:

$\begin{matrix}{{\frac{\left( H_{T} \right)_{new} - \left( H_{T} \right)_{old}}{\left( H_{T} \right)_{new}}} < 10^{- 6}} & {{Eq}.\mspace{14mu} 78}\end{matrix}$

-   5. The solution for the flow and heat transfer inside the two layers    is determined.-   6. Time is advanced by Δτ* and steps (1) to (5) are repeated.

Numerical investigations were performed using different mesh sizes andtime steps to assess and ascertain grid and time step independentresults. Any reduction in the values of Δξ, Δη and Δτ* below Δξ=0.008,Δη=0.03 and Δτ*=0.001 cause less than about 0.2 percent error in theresults.

The maximum value of the parameters P_(S) is chosen to be 1.0. Beyondthis value, the error associated with the low Reynolds number model willincrease for moderate values of the dimensionless thermal expansionparameter, softness index of the seals, and the Prandtl number. As anexample, the order of transient and convective terms in the momentumequations is expected to be less than 5.0 percent that of the diffusiveterms for P_(S)=1.0, Pr=6.7, E₁*=E₂*=0.3, F_(T)=0.15, β_(q)=0.2 andσ₁=3.0, σ₂=6.0. The parameters correspond, for example, to a main thinfilm filled with water and having B=D=60 mm, h_(o)=0.3 mm, ω=1.7 s⁻¹,V_(o)=0.1 m/s and K*=33000 N/m.

3E. Discussions of the Results

Ideal gases produce about a 15 percent increase in the closed cavityvolume under typical room conditions for a 45° C. temperaturedifference. Further, about a 60 percent increase in the convective heattransfer coefficient for about a 2 percent volume fraction of copperultrafine particles has been reported. See Li & Xuan (2002) Science inChina (Series E) 45:408-416, which is herein incorporated by reference.Accordingly, the parameters F_(T) and λ₂ were varied until comparablechanges have been attained in the dimensionless thin film thickness andthe Nusselt number.

3F. Softness Index and Thermal Expansion Parameters of the Seal

FIG. 26 illustrates the effects of the softness index of the seals ofthe present invention on the dynamics and thermal characterizations of atwo-layered thin film operating at the CIP condition. The softness indexwas considered to be equal for both layers, denoted by E* andcorresponds to the case when both lower and upper layers fluids areidentical. As the softness index E* increases, the dimensionless flowrate parameter for the primary fluid layer Ψ_(X=0.5) increases asdescribed by the solid lines displayed in FIG. 26A. This is expected forcases where the average pressure of the lower layer is greater than thatof the upper layer. Meanwhile the disturbance in the primary fluid layerthickness increases as E* increases as depicted by the dotted line shownin FIG. 26B. This phenomenon can be utilized in enhancing the coolingdue to thermal dispersion in the secondary fluid flow as proposed byEquation 53. On the other hand, these disturbances may causemalfunctioning of any sensing devices placed in the flow passage sinceboth flow dynamical effects and chemical reactions will be affected. Theincrease in Ψ_(X=0.5) as E* increases causes an increase in the averagedimensionless heat transfer Θ_(AVG) in the primary fluid layer and anincrease in the average upper substrate temperature (θ_(u))_(AVG) asshown in FIG. 26B due to the shrinkage in the upper layer.

For the CIP condition, the increase in the dimensionless thermalexpansion parameter F_(T) of the upper flexible complex seals causes areduction in Ψ_(X=0.5) values and an increase in the disturbance atintermediate substrate. Consequently, the parameters Θ_(AVG) and(θ_(u))_(AVG) decrease as F_(T) increases. These observations are shownin FIG. 27 which corresponds to a parametric case with water as theprimary fluid while the secondary fluid is taken to be air. For CIFcondition, the compression in the primary fluid layer film thicknessincreases the flow near the lower and intermediate substrates, therebyenhancing the thermal convection as illustrated in FIG. 28. As a result,thermally developed conditions are achieved within shorter distance fromthe inlet as F_(T) increases. This alleviates thermal effects such asbimaterial effects in sensors.

3G. Role of the Squeezing and Thermal Squeezing Parameters

As the squeezing number for the primary fluid flow passage increases,the net pressure force on the intermediate substrate decreases asdictated by Equation 49. Therefore, the primary fluid layer filmthickness decreases causing a reduction in the values of Ψ_(X=)0.5,(Θ_(AVG) and (θ_(u))_(AVG) as shown in FIG. 29. The disturbance at theintermediate substrate, variation in dH₁/dτ, decreases slightly as σ₁increases as shown in FIG. 29A. This phenomenon is ascribed to the factthat the relief in the thickness of the upper layer tend to minimize theeffects of the internal pressure pulsations on the moving substrate. SeeFIG. 26A.

The increase in the value of the thermal squeezing parameter P_(S2) ofthe upper layer causes an enhancement in the upper substrate cooling asshown by reductions in (θ_(u))_(AVG) in FIG. 30B. By introducing saltconcentrations or due to the presence of scales, suspensions as a resultof corrosion in different components or from incomplete combustion, inthe secondary fluid, the value of P_(S2) can be altered, thereby causingan increase in E₂* which can be kept constant by selecting the upperlayer width D₂ using methods known in the art. Due to reductions in(θ_(u))_(AVG) as P_(S2) increases, the upper layer film thicknessdecreases allowing for more flooding in the primary fluid layer. Thus,the average heat transfer in the primary fluid layer increases as P_(S2)increases. See FIG. 30. The variation in dH₁/dτ decreases slightly asP_(S2) increases due to reductions in H_(T) noting that the intermediatesubstrate becomes more stable for the effects that makes it closer toeither the upper or lower substrates for a given softness index. Theincrease in the cooling of the upper layer due to an increase in P_(S2)causes a relief in the primary fluid layer film thickness resulting in areduction in its Nusselt number. See FIG. 31 for the CIF condition.Accordingly, the main inlet temperature is convected further downstreamwhich may increase noise levels due bimaterial effects of certainsensors.

3H. Role of Thermal Dispersion Due to Ultrafine Suspensions

Due to their random motions, ultrafine particles tend to increase theheat exchange within the fluid causing the thermal dispersion effect.Therefore, as the dimensionless thermal dispersion parameter λincreases, the thermal conductivity increases causing the uppersubstrate temperature (θ_(u))_(AVG) to decrease. Thus, in turn, thevalues of Ψ_(X=0.5) and Θ_(AVG) are increased while variations in dH₁/dτare decreased as λ increases. See FIG. 32 for the CIP condition. Assuch, the stability of the intermediate substrate is enhanced in thepresence of dispersive flows. For the CIF condition, the relief in theprimary fluid layer film thickness due to an increase in λ, as shown inFIG. 33A, reduces the convective heat transfer coefficient of theprimary fluid layer. Thus, a decrease in Θ_(AVG) is associated as shownin FIG. 33B.

3I. Role of Pulsation Frequency and Total Thickness of the Two Layers

FIG. 34 shows the effects the frequency of pressure pulsation γ_(p) onfluctuations of Ψ_(X=0.5) and (θ_(u))_(AVG). These fluctuations aredefined as:

$\begin{matrix}{{\Delta\;\Psi_{X = 0.5}} = \frac{\left( \Psi_{X = 0.5} \right)_{\max} - \left( \Psi_{X = 0.5} \right)_{\min}}{2}} & {{{Eq}.\mspace{14mu} 79}a} \\{{\Delta\;\Theta_{AVG}} = \frac{\left( \Theta_{AVG} \right)_{\max} - \left( \Theta_{AVG} \right)_{\min}}{2}} & {{{Eq}.\mspace{14mu} 79}b}\end{matrix}$where the maximum and minimum values corresponds to the steady periodicvalues.

It should be noted that ΔΨ_(X=0.5) and ΔΘ_(AVG) are unpredictable atrelatively lower frequencies of pulsations and the primary fluid layerbecomes more stable for large values of γ_(p). See FIG. 34. FIG. 35shows that the reduction in the primary fluid layer flow rate decreasesas the dimensionless total thickness H_(t) increases. This is becausemore cooling is expected to the upper substrate as H_(t) increasesresulting in reducing the volumetric thermal expansion effects of thestagnant fluid. As such, the fluctuating rate at the intermediatesubstrate is reduced as H_(t) increases for the selected range as shownin FIG. 35.

4. Cooling Enhancements in Thin Films Supported by Flexible ComplexSeals in the Presence of Ultrafine Suspensions

As provided herein, flow and heat transfer inside thin films supportedby flexible complex seals, flexible seals having closed cavities of astagnant fluid possessing a large coefficient of volumetric thermalexpansion β_(T), were studied in the presence of suspended ultrafineparticles and under periodically varying thermal load conditions. Thegoverning continuity, momentum and energy equations arenon-dimensionalized and reduced to simpler forms. The deformation of theseal is related to the internal pressure and lower substrate'stemperature based on the theory of linear elasticity and a linearizedmodel for thermal expansion. As provided herein, enhancements in thecooling may be achieved by an increase in the volumetric thermalexpansion coefficient, thermal load, thermal dispersion effects,softness of the supporting seals and the thermal capacitance of thecoolant fluid. Further, thermal dispersion effects were found toincrease the stability of the thin film. The noise in the thermal loadwas found to affect the amplitude of the thin film thickness, Nusseltnumber and the lower substrate temperature; however, it had a negligibleeffect on the mean values.

Thin films are widely used in cooling of many heating sources such aselectronic components. These elements are used in thin films in coolingsystems such as in flat heat pipes or microchannel heat sinks. See Moonet al. (2000) Int. J. Microcircuits and Electronic Packaging 23:488-493,Fedorov & Viskanta (2000) Int. J. Heat and Mass Transfer 43:399-415, andZhu & Vafai (1999) Int. J. Heat and Mass Transfer 42:2287-2297, whichare herein incorporated by reference. A two phase flow in microchannelis capable of removing maximum heat fluxes generated by electronicpackages yet the system may become unstable near certain operatingconditions. See Bowers & Mudawar (1994) ASME J. Electronic Packaging116:290-305, which is herein incorporated by reference. Further, the useof porous medium in cooling of electronic devices was found to enhanceheat transfer due to increases in the effective surface area. See Hadim(1994) ASME J. Heat Transfer 116:465-472, which is herein incorporatedby reference. However, the porous medium creates a substantial increasein the pressure drop inside the thin film.

As provided herein, additional cooling can be achieved if the thin filmthickness is allowed to increase by an increase in the thermal loadwhich will cause the coolant flow rate to increase using flexiblecomplex seals of the present invention, i.e. flexible seals havingclosed cavities of a stagnant fluid having a large value of thevolumetric thermal expansion coefficient β_(T).

In the presence of periodic external thermal loads, the thickness of athin film supported by a flexible complex seal is expected to beperiodic. This is because the stagnant fluid expands during maximumthermal load intervals allowing for a relaxation in the thin filmthickness which causes a flooding of the coolant. On the other hand, thethin film is squeezed during minimum thermal loads intervals due to thecontraction in the stagnant fluid in the closed cavities of the flexiblecomplex seals.

One of the advantages of using flexible complex seals is that theincrease in the coolant flow rate because of thermal expansion effectsproduces an additional cooling in the presence of suspended ultrafineparticles. See Li & Xuan (2002) Science in China (Series E) 45:408-416,which is herein incorporated by reference. This is because the chaoticmovement of the ultrafine particles, the thermal dispersion, increaseswith the flow where it is modeled in the energy equation by introducingan effective thermal conductivity of the coolant. See Xuan & Roetzel(2000) Int. J. Heat and Mass Transfer 43:3701-3707, which is hereinincorporated by reference. Further, large fluctuation rates that can begenerated in the flow during severe squeezing conditions tend toincrease the chaotic motions of the particles in the fluid whichincreases the energy transport in the coolant.

As provided herein, the enhancement in the cooling process inside thinfilms supported by flexible complex seals in the presence of suspendedultrafine particles was analyzed. The lower substrate of the examinedthin film was considered to be under a periodically varying heat flux.The thin film thickness was related to the thermal load and the internalpressure through the volumetric thermal expansion coefficient of thestagnant fluid and the theory of linear elasticity applied to thesupporting seals. The governing equations for flow and heat transferwere properly non-dimensionalized and reduced into simpler equations forlow Reynolds numbers. The resulting equations were then solvednumerically to determine the effects of the thermal load, volumetricthermal expansion coefficient of the stagnant fluid, the softness of theseal, thermal capacitance of the working fluid and the squeezing numberon the dynamics and thermal characteristic of the thin films supportedby flexible complex thin films. As provided herein, the flexible complexseals of the present invention are useful in enhancing the cooling andcan be used for additional purposes such as for diagnosing functions forheating sources so long as they possess large thermal expansioncoefficient.

The following Table 7 provides the various symbols and meanings used inthis section:

TABLE 7 A* a closed cavity dimension parameter B thin film length C_(F)volumetric thermal expansion correction factor C* coefficient of thermaldispersion c_(p) average specific heat of the working fluid or thedilute mixture D width of the thin film d_(s) characteristic parameterof the seal E effective modulus of elasticity for the sealing assembly Gwidth of the closed cavity F_(n) fixation parameter F_(T) dimensionlessthermal expansion parameter H, h, h_(o) dimensionless, dimensional andreference thin film thicknesses h_(c) convective heat transfercoefficient k thermal conductivity of the working fluid or the dilutemixture k_(o) reference thermal conductivity of the fluid Nu_(L) lowersubstrate's Nusselt number P_(S) thermal squeezing parameter p fluidpressure q_(o) reference heat flux at the lower substrate T, T₁temperature in fluid and the inlet temperature t Time V_(o) referenceaxial velocity U, u dimensionless and dimensional axial velocities V, vdimensionless and dimensional normal velocities X, x dimensionless anddimensional axial coordinates Y, y dimensionless and dimensional normalcoordinates β_(q) dimensionless amplitude of the thermal load β_(T)coefficient of volumetric thermal expansion of the stagnant fluid εperturbation parameter γ dimensionless frequency μ averaged dynamicviscosity of the working fluid or the dilute mixture θ, θ_(m)dimensionless temperature and dimensionless mean bulk temperature θ_(W)dimensionless temperature at the lower substrate ρ averaged density ofthe working fluid or the dilute mixture υ averaged kinematic viscosityof the working fluid or the dilute mixture τ, τ* dimensionless time σsqueezing number ω reciprocal of a reference time (reference squeezingfrequency) η variable transformation for the dimensionless Y-coordinateλ dimensionless thermal dispersion parameter Π dimensionless pressureΠ_(i) dimensionless inlet pressure Λ reference lateral to normalvelocity ratio4A. Problem Formulation

FIG. 36 shows a thin film having a flexible complex seal. The flexiblecomplex seal contains closed cavities filled with a stagnant fluidhaving relatively a large coefficient of volumetric thermal expansion.Flexible seals are also included in order to allow the thin film toexpand. The flexible seals and flexible complex seals of the presentinvention may comprise a closed cell rubber foam. See Friis et al.(1988) J. Materials Science 23:4406-4414, which is herein incorporatedby reference. Any excessive heat increases the temperature of thesubstrate. Thus, the stagnant fluid becomes warmer and expands. Theflexible seals are flexible enough so that the expansion results in anincrease in the separation between the lower and the upper substrates.Accordingly, the flow resistance of the working fluid passage decreasescausing a flooding of the coolant. As a result, the excessive heatingfrom the source is removed. The flexible seals can be placed betweenguiders, as shown in FIG. 36B, in order to minimize side expansion ofthe seals and maximize transverse thin film thickness expansion.

The analysis is concerned with a thin film that has a small thickness hcompared to its length B and its width D. Therefore, a two-dimensionalflow is assumed. The x-axis was taken along the axial direction of thethin film while y-axis was taken along its thickness as shown in FIG.36A. Further, the film thickness was assumed to be independent of theaxial coordinate such as in two main cases: symmetric thin films havinga fluid injected from the center as shown in FIG. 36C and in multiplepassages thin films having alternating coolant flow directions.

The lower substrate of the thin film was assumed to be fixed (immobileand inflexible substrate) and in contact with or adjacent to a heatingsource while the upper substrate was attached to the lower substrate byflexible complex seals allowing it to expand (mobile and inflexiblesubstrate). The motion of the upper substrate due to both internalvariations in the stagnant fluid temperature and the induced internalpressure pulsations as a result of oscillating thermal loads isexpressed according to the following relation:

$\begin{matrix}{{H \equiv \frac{h}{h_{o}}} = \left( {1 + H_{T} + H_{p}} \right)} & {{Eq}.\mspace{14mu} 80}\end{matrix}$where

h is the thin film thickness

h_(o) is a reference film thickness

H is the dimensionless thin film thickness

H_(T) is the dimensionless motion of the upper substrate due to thethermal expansion of the stagnant fluid

H_(p) is the dimensionless motion of the upper substrate as a result ofthe deformation of seals due to the average internal pressure of theworking fluid

The fluid is assumed to be Newtonian having constant average propertiesexcept for the thermal conductivity. The general two-dimensionalcontinuity, momentum and energy equations for a laminar flow of theworking fluid inside the thin film are given as:

$\begin{matrix}{{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}} = 0} & {{Eq}.\mspace{20mu} 81} \\{{\rho\left( {\frac{\partial u}{\partial t} + {u\frac{\partial u}{\partial x}} + {v\frac{\partial u}{\partial y}}} \right)} = {{- \frac{\partial p}{\partial x}} + {\mu\left( {\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}}} \right)}}} & {{Eq}.\mspace{14mu} 82} \\{{\rho\left( {\frac{\partial v}{\partial t} + {u\frac{\partial v}{\partial x}} + {v\frac{\partial v}{\partial y}}} \right)} = {{- \frac{\partial p}{\partial y}} + {\mu\left( {\frac{\partial^{2}v}{\partial x^{2}} + \frac{\partial^{2}v}{\partial y^{2}}} \right)}}} & {{Eq}.\mspace{14mu} 83} \\{{\rho\;{c_{p}\left( {\frac{\partial T}{\partial t} + {u\frac{\partial T}{\partial x}} + {v\frac{\partial T}{\partial y}}} \right)}} = {{\frac{\partial}{\partial x}\left( {k\frac{\partial T}{\partial x}} \right)} + {\frac{\partial}{\partial y}\left( {k\frac{\partial T}{\partial y}} \right)}}} & {{Eq}.\mspace{14mu} 84}\end{matrix}$where

T is temperature

u is the dimensional axial velocity

v is the dimensional normal velocity

ρ is the average density

p is pressure

μ is the average dynamic viscosity

c_(p) is the average specific heat

k is the thermal conductivity

The previous fluid properties are for the pure working fluid in the casewhere the fluid is free from any suspensions. In the presence ofsuspended ultrafine particles, the previous properties will be for anapproximated new continuum fluid composed from the mixture of the purefluid and the suspensions. See Xuan & Roetzel (2000) Int. J. Heat andMass Transfer 43:3701-3707, which is herein incorporated by reference.The new properties of the mixture are related to the fluid and theparticle properties through the volume fraction of the suspendedparticles inside the thin film and the thermal dispersion parameter.

The following dimensionless variables were used to non-dimensionalizedEquations 81-84:

$\begin{matrix}{X = \frac{x}{B}} & {{{Eq}.\mspace{14mu} 85}a} \\{Y = \frac{y}{h_{o}}} & {{{Eq}.\mspace{14mu} 85}b} \\{\tau = {\omega\; t}} & {{{Eq}.\mspace{14mu} 85}c} \\{U = \frac{u}{\left( {{\omega\; B} + V_{o}} \right)}} & {{{Eq}.\mspace{14mu} 85}d} \\{V = \frac{v}{h_{o}\omega}} & {{{Eq}.\mspace{14mu} 85}e} \\{\Pi = \frac{p - p_{e}}{{\mu\left( {\omega + \frac{V_{o}}{B}} \right)}ɛ^{- 2}}} & {{{Eq}.\mspace{14mu} 85}f} \\{\theta = \frac{T - T_{1}}{\left( {q_{o}h_{o}} \right)/k_{o}}} & {{{Eq}.\mspace{14mu} 85}g}\end{matrix}$where ω, T₁, p_(e), q_(o) and V_(o) are the reference frequency ofthermal load, inlet temperature of the fluid, a constant representingthe exit pressure, reference heat flux and a constant representing areference dimensional velocity, respectively. The term k_(o) correspondsto the working fluid thermal conductivity in the absence of anysuspensions while it is the stagnant thermal conductivity, free from thedispersion term, for the dilute mixture between the fluid and theultrafine suspensions. The stagnant thermal conductivity has usually anenhanced value when compared to that of the pure fluid for metallicparticles. See Eastman et al. (2001) Applied Physics Letters 78:718-720,which is herein incorporated by reference.

The upper substrate is assumed to be insulated to simplify the analysisand that the lower substrate was subjected to a periodically varyingwall heat flux q_(L) condition according to the following relation:q _(L) =q _(o)(1+β_(q) sin(γωt))  Eq. 86where β_(q) and γ are the dimensionless amplitude of the lowersubstrate's heat flux and a dimensionless frequency, respectively. Thevariables X, Y, τ, U, V, Π and θ are the dimensionless forms of x, y, t,u, v, p and T variables, respectively. The parameter εappearing inEquation 85f is the perturbation parameter,

$ɛ = {\frac{h_{o}}{B}.}$

For the thin film shown in FIG. 36C, the displacement of the uppersubstrate due to internal pressure variations is related to the averagedimensionless pressure of the working fluid, Π_(AVG), through the theoryof linear elasticity by the following relation:H_(p)=F_(n)Π_(AVG)  Eq. 87

This is based on the fact that the upper substrate is assumed to berigid and that the applied force on an elastic material, the flexibleseal, is assumed to behave as an elastic material, is proportional tothe elongation of this material. See Norton (1998) MACHINE DESIGN; ANINTEGRATED APPROACH Prentice-Hall, New Jersey, which is hereinincorporated by reference. The parameter F_(n) is referred to as thefixation parameter and is a measure of the softness of the seal,flexible seals have large F_(n) values, and is equal to:

$\begin{matrix}{F_{n} = \frac{\mu\left( {V_{o} + {\omega\; B}} \right)}{E\; ɛ^{2}d_{s}}} & {{Eq}.\mspace{14mu} 88}\end{matrix}$where E and d_(s) are the effective modulus of elasticity for thecomplex seal and a characteristic parameter which depends on the seal'sdimensions and the thin film width D, respectively. The quantity d_(s)is equal to the effective dimension of the seal's cross section timesthe ratio of the total length of the seal divided by the thin film widthD. The seal is considered to have isotropic properties. Further, theeffective dimension of the seals times their total length represents thecontact area between the seals and the upper or lower substrates whenthe seals have a rectangular cross section as shown in FIG. 36. Otherthan this, the effective diameter requires a theoretical determination.

As provided herein, the analysis was performed for relatively smallthermal load frequencies in order to ascertain that squeezing generatedflows have relatively small Reynolds numbers. For these frequencies,Equation 87 is applicable and the inertia effect of the upper substrateis negligible. Moreover, the increase in the thickness due to a pressureincrease in the thin film causes a reduction in the stagnant fluidpressure. This action stiffens the insulating assembly. Therefore, theparameter E is considered to be the effective modulus of elasticity forthe insulating assembly not for the seal itself. Practically, the closedcavity width G is assumed to be large enough such that a small increasein the stagnant fluid pressure due to the expansion can support theassociated increase in the elastic force on the seal.

The dimensionless displacement of the upper substrate due to thermalexpansion is related to the dimensionless average temperature of thelower substrate, (θ_(W))_(AVG), by the following linearized model:H _(T) =F _(T)(θ_(W))_(AVG)  Eq. 89where F_(T) is named the dimensionless thermal expansion parameter andis equal to:

$\begin{matrix}{F_{T} = {\frac{A^{*}\beta_{T}q_{o}h_{o}}{k_{o}}C_{F}}} & {{Eq}.\mspace{14mu} 90}\end{matrix}$where A* is a constant depending on the closed cavities dimensions andgeometry. The parameter β_(T) is the volumetric thermal expansioncoefficient of the stagnant fluid in its approximate form:β_(T)≈(1/V_(So))[(V_(S)−V_(S1))/(T_(S)−T₁)]|_(p) _(S1) evaluated at thepressure p_(s1) corresponding to the stagnant fluid pressure at theinlet temperature T₁. The quantities V_(S1) and V_(S) represent theclosed cavity volumes at normal operating conditions when the stagnantfluid is at T₁ and at the present stagnant fluid temperature T_(S),respectively. The parameter V_(So) represents the closed cavity volumeat the reference condition. The factor C_(F) represents the volumetricthermal expansion correction factor. This factor was introduced in orderto account for the increase in the stagnant pressure due to the increasein the elastic force in the seal during the expansion which tends todecrease the effective volumetric thermal expansion coefficient. Itapproaches one as the closed cavity width G increases and may bedetermined theoretically using methods known in the art.

The parameter F_(T) is enhanced at elevated temperatures for liquids andat lower temperature for gases because β_(T) increases for liquids anddecreases for gases as the stagnant temperature increases. Dimensionlessthermal expansion parameter is also enhanced by a decrease in k_(o), anincrease in q_(o), an increase in F_(n) or by increases in h_(o).Equation 89 is based on the assumption that the stagnant fluidtemperature is similar to the lower substrate temperature since entireclosed cavity surfaces were considered insulated except that facing thelower substrate. Furthermore, the heat flux of the heating source isapplied on the portion of the lower substrate that is facing the workingfluid. The other portion which faces the seals is taken to be isolatedfrom the heating source and the environment to minimize the variation inthe lower substrate temperature along the width direction.

In the presence of suspended ultrafine particles in the working fluid,the thermal conductivity of the working fluid composed from the purefluid and suspensions is expected to vary due to the thermal dispersion.To account for these variations, the following model which is similar tothe Xuan & Roetzel ((2000) Int. J. Heat and Mass Transfer 43:3701-3707)model that linearly relates the effective thermal conductivity of theworking fluid to the fluid speed is utilized:k(X, Y, τ)=k _(o)(1+λ√{square root over (U ²(X, Y, τ)+Λ² V ²(X, Y,τ))}{square root over (U ²(X, Y, τ)+Λ² V ²(X, Y, τ))})=k _(o)φ(X, Y,τ)  Eq. 91where λ and Λ are the dimensionless thermal dispersion coefficient andthe reference squeezing to lateral velocity ratio which are:

$\begin{matrix}{\lambda = {{C^{*}\left( {\rho\; c_{p}} \right)}{h_{o}\left( {V_{o} + {\omega\; B}} \right)}}} & {{{Eq}.\mspace{14mu} 92}a} \\{\Lambda = \frac{ɛ\sigma}{12}} & {{{Eq}.\mspace{14mu} 92}b}\end{matrix}$where C* is the coefficient of the thermal dispersion which depends onthe diameter of the ultrafine particles, its volume fraction (ratio ofthe particles volume to the total thin film volume), and both fluid andultrafine particles properties. Ultrafine particles include particlesthat are extremely small compared with the thickness of the thin film.

The coefficient C* is expected to increase by an increase in thediameter of the particles, their volume fraction, their surfaceroughness and the working fluid Prandtl number, Pr=(ρc_(p)ν)/k_(o). Onthe other hand, the stagnant thermal conductivity k_(o) increases withan increase in both the volume fraction and the surface area of theparticles. A dilute mixture of ultrafine suspensions and water produceno significant change in the pressure drop compared to pure water whichreveals that the viscosity is a weak function of the fluid dispersionfor a dilute mixture.

Generally, flows inside thin films are in laminar regime and could becreep flows as in lubrication. Therefore, the low Reynolds numbers (themodified lateral Reynolds number Re_(L)=(V_(o)h_(o))ε/ν and thesqueezing Reynolds number Re_(S)=(h_(o) ²ω)/ν) flow model was usedherein. These insulating assemblies neglect the transient and convectiveterms in momentum equations, Equations 82 and 83. These terms becomeincomparable to the pressure gradient and diffusive terms for smallsqueezing frequencies and reference velocities. Application of theseinsulating assemblies to Equations 82-84 and the outcome ofdimensionalizing the energy equation, Equation 85, result in thefollowing reduced non-dimensionalized equations:

$\begin{matrix}{U = {\frac{1}{2}\frac{\partial\Pi}{\partial X}{H^{2}\left( \frac{Y}{H} \right)}\left( {\frac{Y}{H} - 1} \right)}} & {{Eq}.\mspace{14mu} 93} \\{V = {\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {{3\left( \frac{Y}{H} \right)^{2}} - {2\left( \frac{Y}{H} \right)^{3}}} \right)}} & {{Eq}.\mspace{14mu} 94} \\{{\frac{\partial}{\partial X}\left( {H^{3}\frac{\partial\Pi}{\partial X}} \right)} = {\sigma\frac{\partial\Pi}{\partial\tau}}} & {{Eq}.\mspace{14mu} 95} \\{{P_{S}\left( {\frac{\partial\theta}{\partial\tau} + {\frac{12}{\sigma}U\frac{\partial\theta}{\partial X}} + {V\frac{\partial\theta}{\partial Y}}} \right)} = {\frac{\partial}{\partial Y}\left( {\phi\frac{\partial\theta}{\partial Y}} \right)}} & {{Eq}.\mspace{14mu} 96}\end{matrix}$

Note that Equation 96 is based on the assumption that the axialconduction is negligible when compared to the transverse conduction. Theparameters σ and P_(S) are referred to as the squeezing number and thethermal squeezing parameter, respectively, and are defined as:

$\begin{matrix}{\sigma = \frac{12}{1 + \frac{V_{o}}{\omega\; B}}} & {{{Eq}.\mspace{14mu} 97}a} \\{P_{S} = \frac{\rho\; c_{p}h_{o}^{2}\omega}{k_{o}}} & {{{Eq}.\mspace{14mu} 97}b}\end{matrix}$

Both inlet and exit dimensionless pressures were assumed constant andthe following relationship was obtained between the inlet dimensionlesspressure and the squeezing number based on the assumption that thereference velocity V_(o) represents the average velocity in the thinfilm at zero values of F_(T) and F_(n):Π_(i)=12−σ  Eq. 98

Accordingly, the dimensionless pressure gradient, the dimensionlesspressure and the average dimensionless pressure Π_(AVG) inside the thinfilm are related to the squeezing number through the followingequations:

$\begin{matrix}{\frac{\partial{\Pi\left( {X,\tau} \right)}}{\partial X} = {{\frac{\sigma}{H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {X - \frac{1}{2}} \right)} - \left( {12 - \sigma} \right)}} & {{Eq}.\mspace{14mu} 99} \\{{\Pi\left( {X,\tau} \right)} = {{\frac{\sigma}{2H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {X^{2} - X} \right)} - {\left( {12 - \sigma} \right)\left( {X - 1} \right)}}} & {{Eq}.\mspace{14mu} 100} \\{{\Pi_{AVG}(\tau)} = {{{- \frac{\sigma}{12H^{3}}}\frac{\mathbb{d}H}{\mathbb{d}\tau}} + \frac{\left( {12 - \sigma} \right)}{2}}} & {{Eq}.\mspace{14mu} 101}\end{matrix}$

The dimensionless thermal boundary conditions for the previously definedproblem are taken as follows:

$\begin{matrix}{{{\theta\left( {X,Y,0} \right)} = 0},{{\theta\left( {0,Y,\tau} \right)} = 0},{\frac{\partial{\theta\left( {X,0,\tau} \right)}}{\partial Y} = {- \left( {1 + {\beta_{q}{\sin({\gamma\tau})}}} \right)}},{\frac{\partial{\theta\left( {X,H,\tau} \right)}}{\partial Y} = 0}} & {{Eq}.\mspace{14mu} 102}\end{matrix}$

Based on the physical conditions, the Nusselt number is defined as:

$\begin{matrix}\begin{matrix}{{{Nu}_{L}\left( {X,\tau} \right)} \equiv \frac{h_{c}h_{o}}{k}} \\{= \frac{1}{{\theta\left( {X,0,\tau} \right)} - {\theta_{m}\left( {X,\tau} \right)}}} \\{= \frac{1}{{\theta_{w}\left( {X,\tau} \right)} - {\theta_{m}\left( {X,\tau} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} 103}\end{matrix}$

The parameter θ_(m) is the dimensionless mean bulk temperature and isgiven as:

$\begin{matrix}{{{\theta_{m}\left( {X,\tau} \right)} = {\frac{1}{{U_{m}\left( {X,\tau} \right)}H}{\int_{0}^{H}{{U\left( {X,Y,\tau} \right)}{\theta\left( {X,Y,\tau} \right)}\ {\mathbb{d}Y}}}}}{{U_{m}\left( {X,\tau} \right)} = {\frac{1}{H}{\int_{0}^{H}{{U\left( {X,Y,\tau} \right)}\ {\mathbb{d}Y}}}}}} & {{Eq}.\mspace{14mu} 104}\end{matrix}$where U_(m) is the dimensionless average velocity at a given section.4B. Numerical Procedure

The procedure for the numerical solution is summarized as follows:

-   1. Initially, a value for H_(T) is assumed.-   2. At the present time, the dimensionless thickness of the thin film    H is determined by solving Equations 80, 87, 88, and 101    simultaneously, using an explicit formulation. The velocity field, U    and V, is then determined from Equations 93, 94, and 99.-   3. At the present time, the reduced energy equation, Equation 96, is    transferred into one with constant boundaries using the following    transformations:

${\tau^{*} = \tau},{\xi = {{X\mspace{14mu}{and}\mspace{14mu}\eta} = {\frac{Y}{H}.}}}$A tri-diagonal solution was implemented along with a marching scheme.See Blottner (1970) AIAA J. 8:193-205, which is herein incorporated byreference. Backward differencing was chosen for the axial convective andtransient terms and central differencing was selected for thederivatives with respect to η. The values of 0.008, 0.03, 0.001 werechosen for Δξ, Δη and Δτ*, respectively.

-   4. H_(T) is updated from Equation 89 and steps (2) to (4) is    repeated until:

$\begin{matrix}{{\frac{\left( H_{T} \right)_{new} - \left( H_{T} \right)_{old}}{\left( H_{T} \right)_{new}}} < 10^{- 6}} & {{Eq}.\mspace{14mu} 105}\end{matrix}$

-   5. The converged solution for the flow and heat transfer inside the    thin film is determined at the present time.-   6. Time is advanced by Δτ* and steps (1) to (5) are repeated.

Numerical investigations were performed using different mesh sizes andtime steps to assess and ascertain grid and time step independentresults. Any reduction in the values of Δξ, Δη and Δτ* below Δξ=0.008,Δη=0.03 and Δτ*=0.001 results in less than about a 0.2 percent error inthe results.

In the results, the maximum value of the parameters P_(S) is chosen tobe 1.0. Beyond this value, the error associated with the low Reynoldsnumber model will increase for moderate values of the dimensionlessthermal expansion parameter, fixation parameter, and the Prandtl number.As an example, the order of transient and convective terms in themomentum equations were found to be less 1.0 percent that of thediffusive terms for P_(S)=1.0, Pr=6.0, F_(n)=0.05, F_(T)=0.25, β_(q)=0.1and σ=6.0. The parameters correspond, for example, to a thin film filledwith water and having B=D=60 mm, h_(o)=0.3 mm, d_(s)=0.5 mm, ω=2.0 s⁻¹,V_(o)=0.12 m/s and E=2(10⁵) pa.

4C. Discussions of the Results

Ideal gases produce about a 15 percent increase in the closed cavityvolume at room conditions for a 45° C. maximum temperature difference.Further, about a 60 percent increase in the convective heat transfercoefficient for a volume fraction of copper ultrafine particles of abouta 2.0 percent has been reported. See Li & Xuan (2002) Science in China(Series E) 45:408-416, which is herein incorporated by reference.Accordingly, the parameters F_(T) and λ were varied until comparablechanges have been attained in the dimensionless thin film thickness andthe Nusselt number.

4D. Effects of Dimensionless Thermal Expansion Parameter

FIG. 37A illustrates the effects of the dimensionless thermal expansionparameter F_(T) on the dimensionless thickness H of the thin film. Theparameter F_(T) can be increased either by an increase in the volumetricthermal expansion coefficient of the stagnant fluid or by an increase indimensional reference temperature (q_(o)h_(o))/k_(o). Both factors makethe flexible complex seal softer. Thus, dimensionless thickness H isincreased as F_(T) increases as shown in FIG. 37A. This allows morecoolant to flow causing reductions in the average dimensionless lowersubstrate's temperature (θ_(W))_(AVG) as clearly seen in FIG. 37B whichcan provide additional cooling to any heated surface such as surfaces ofelectronic components.

FIG. 37B also indicates that as thermal load increases, the averagelower substrate's temperature increases; however, this increase can bereduced by using a flexible complex seal. This additional cooling may beobtained with no need for external controlling devices, therebyproviding extra safety for an electronic components such as a heatedsurface, when the thermal loads increase over the projected capacity.The fluctuation rate at the upper substrate, |dH/dτ|, is noticed toincrease as F_(T) increases as shown in FIG. 37C, which may be anadvantage for the cooling process especially at high levels offluctuation rates the thermal dispersion will be enhanced in the coolantwhen suspended ultrafine particles are present. The Nusselt number isdecreased as F_(T) increases as shown in FIG. 37D because it isinversely proportional to H, which is the reason that the percentagedecrease in lower substrate temperatures is lower than the percentageincrease in the thin film thickness as F_(T) increases.

4E. Effects of Dimensionless Thermal Dispersion Parameter

FIG. 38A describes the effects of the dimensionless thermal dispersionparameter λ of the coolant fluid on the average lower substrate'stemperature of the thin film. This parameter can be increased either byincreasing the diameter of the ultrafine particles or increasing theroughness of these particles while keeping a fixed volume fractioninside the coolant, thereby ensuring that thermal squeezing parameterremains constant. FIG. 38A shows that the thermal dispersion can provideadditional cooling to a heated element, thereby causing an additionalreduction in the average dimensionless lower substrate temperature(θ_(W))_(AVG). Part of this cooling is due to the expansion processsince it results in flooding of the working fluid which increases theirregularity and the random motion of the particles. This causesadditional enhancements in the energy exchange rate. Another part forthe enhancement in the cooling is attributed to the fact that the noisein the thermal load, especially those having heterogeneous fluctuationrates, produces additional squeezing due to the velocities that appearin Equation 91.

Due to the reduction in the lower substrates temperatures as λincreases, the dimensionless thin film thickness decreases as λincreases as depicted in FIG. 38B.

Additional enhancements in the thermal dispersion effect are expected asboth the perturbation parameter and the squeezing number increase assuggested by Equations 91 and 92. Both effects result in a magnificationin the fluctuation rates in the flow which causes additional increasesin the cooling process. As provided herein, the perturbation parameterand the fluctuation rates are small and their effects are notnoticeable.

The reduction in thermal resistance across the transverse direction whenλ increases causes the temperature profiles to be more flattened as λincreases as seen in FIG. 38C. Accordingly, the Nusselt number increasesas λ increases as seen in FIG. 38D. FIG. 39 shows that the fluctuationrate at the upper substrate, |dH/dτ|, decreases as λ increases. As aresult, ultrafine particle suspensions inside thin films supported byflexible complex seals not only cause enhancements in heat transfer butalso make these thin films dynamically more stable. In this insulatingassembly, an increase in λ between zero and unity cause a reduction inthe average lower temperature by dimensionless temperature of about 0.12and an increase in the Nusselt number by about 50 percent.

4F. Effects of Thermal Squeezing Parameter and the Squeezing Number

FIG. 40A shows the effects of the thermal squeezing parameter P_(S) andthe squeezing number σ on the average dimensionless lower substratetemperature (θ_(W))_(AVG). The lower substrate temperature decreases asP_(S) increases and as σ decreases. Both effects tend to increasethermal convection which decreases the lower substrate temperature. Theincrease in P_(S) results in an increase in the thermal capacitance ofthe working fluid and a decrease in σ indicates an increase in thereference velocity. Accordingly, the dimensionless thickness H decreasesas P_(S) increases as shown in FIG. 40B. In addition, the pressure forceinside the thin film increases as σ decreases, thereby causing anincrease in H_(p) while H_(T) decreases as a decreases due to theenhancement in the cooling. As a result, the thin film thickness variesslightly when σ decreases as illustrated in FIG. 40B. As seen in FIG.40C, the fluctuation rate at the upper substrate increases as σincreases while it decreases as P_(S) increases. Also, the fluctuationrate at the upper substrate is shown to be more affected by P_(S) ascompared to σ.

4G. Effects of the Fixation Parameter and the Amplitude of the ThermalLoad

FIG. 41A shows the effects of the fixation parameter F_(n) and thedimensionless amplitude of the thermal load β_(q) on the averagedimensionless lower substrate temperature (θ_(W))_(AVG). Since flexibleseals possess large F_(n) values, H increases and lower substratetemperature decreases as F_(n) increases as shown in FIG. 41A and FIG.41B. Further, these figures show that an increase in the amplitude ofthe heat flux results in an increase in the fluctuation rate at theupper substrate and the lower substrate temperature but their mean(average) values are almost unaffected.

4H. Effects of Dimensionless Thermal Expansion Parameter on AveragePressure

FIG. 42 shows the effects of F_(T) on the average dimensionless pressureinside a thin film supported by a flexible complex seal. The periodicbehavior of the heat flux results in a periodic variation in the averagepressure inside the thin film. The fluctuation in the pressure increasesas F_(T) increases as seen in FIG. 42. Further, the thermal loadexceeding the internal pressure by a phase shift approximately equal toπ/(2γ). According to FIG. 42, the induced pressure pulsation can be usedas a measurable quantity in order to read, diagnose or for feedback tocontrol the heating source.

5. Oscillatory Flow Disturbances and Thermal Characteristics InsideFluidic Cells Due to Fluid Leakage and Wall Slip Conditions

As provided herein, the effects of both fluid leakage and wall slipconditions were studied analytically and numerically on the fluctuationrate in the flow inside non-isothermal disturbed thin films supported byflexible seals within a fluidic cell. Flow disturbances due to internalpressure pulsations and external squeezing are considered in this work.The main controlling parameters are the dimensionless leakage parameter,softness of the seal, squeezing number, dimensionless slip parameter,the thermal squeezing parameter and the power law index. Accordingly,their influences on the fluctuation rate and heat transfercharacteristics inside disturbed thin films were determined. As providedherein, an increase in the dimensionless leakage parameter, softness ofthe seal-upper substrate assembly and the wall slip parameter result inmore cooling and an increase in the fluctuation level in the flow.However, an increase in the squeezing number and the fluid power indexdecreases decrease flow fluctuations.

Thin films are utilized in various chemical and biological systems suchas in biosensing devices. See Lavrik et al. (2001) BiomedicalMicrodevices 3(1):35-44, which is herein incorporated by reference.These biosensing devices have the advantage to accurately, quickly, andeconomically screen patients for the presence of various diseases or canbe used to detect many bio-warfare agents. Many biosensors in the artcomprise at least one microcantilever, wherein detection of a desiredagent is based on the deflection of the free end of the microcantileverthat is caused by the imposed stresses at least one of its surfaces. SeeU.S. patent application Ser. No. 10/422,776, filed 25 Apr. 2003, whichis herein incorporated by reference. This surface stress is due to thereaction, interaction, or binding between a given agent in a fluidsample inside the thin film and a second agent, such as a receptor, thatreacts, interacts, or binds with the given agent, that is immobilized onthe surface of the microcantilever.

Examples of reactions in biomolecular (receptor/analyte) applicationswhich occur within a fluidic cell include: antibody-antigen(receptor/analyte) bindings or DNA hybridization of a pair of DNAstrands (receptor/analyte) having complementary sequences, and the like.An example of antibody-antigen bindings includes the binding ofpolyclonal anti-PSA (prostate-specific antigen) antibody and free PSA(fPSA). See Wu et al. (2001) Nature Biotechnology 19:856-860, which isherein incorporated by reference. In many cases, disturbances exist inthe flow which can disturb the deflection of the microcantilever andproduce a noise in the measurement. See Fritz et al. (2000) Science288:316-318, which is herein incorporated by reference.

Part of the noise in the measurement is due to the fact thatoscillations in the flow may produce an oscillatory drag force on themicrocantilever surface causing it to vibrate. For example, a 100 nmdeflection in the microcantilever due to initial flow disturbances inthe fluidic cell while the microcantilever deflection due toreceptor/analyte binding was of the order of 10 nm has been reported.See Fritz et al. (2000) Science 288:316-318, which is hereinincorporated by reference.

Meanwhile, flow oscillations may change the microcantilever temperaturecausing it to produce an additional noise where the microcantilever iscomposed of two layers (bimaterial) having different coefficients ofthermal expansion. For example, microcantilevers having a 50 nmdeflection due to bimetallic effects, which was five times themicrocantilever deflection due receptor/analyte binding, has beenreported. See Fritz et al. (2000) Science 288:316-318, which is hereinincorporated by reference. The rate of receptor/analyte binding changeswith the flow velocity has been demonstrated. See Prichard et al. (1995)J. Biomechanics 28:1459-1469, which is herein incorporated by reference.As flow oscillations add extra noise due to surface stresses,minimizations of flow oscillations in fluidic cells are desired and maybe achieved according to the present invention.

Flow disturbances can be due to external disturbances or due to internalpressure pulsations when the pumping process is irregular. Even a smallchange in the internal pressure of the fluidic cell can have asubstantial impact since the thickness of the thin film is very small.The impact is more pronounced if the thin film is supported by flexibleseals. Accordingly, the dynamics and thermal characterization of thinfilms will be altered producing a noise in the biosensor measurementwhich is proportional to the fluctuation rate in the flow. Anothersource for flow disturbances is the flow leakage which can seriouslyaffect the operation of the microcantilever. See Raiteri et al. (2000)Electrochimica Acta 46:157-163, which is herein incorporated byreference.

Flow inside oscillatory disturbed thin films has been studied. SeeLanglois (1962) Quarterly of Applied Math. XX:131-150, which is hereinincorporated by reference. Laminar pulsating flows has been studied aswell as effects of internal pressure pulsations on oscillatory squeezedlaminar flow and heat transfer inside thin films supported by flexibleseals. See Hemida et al. (2002) Int. J. Heat and Mass Transfer45:1767-1780 and Khaled & Vafai (2002) Int. J. Heat and Mass transfer45:5107-5115, which are herein incorporated by reference. Unfortunately,the prior art fails to analyze the effects of fluid leakage on pulsatingflow and heat transfer inside thin films in the presence of flexibleseals. Such an analysis and understanding is important as the effects offluid leakage contributes to flow disturbances.

Thus, as provided herein the effects of fluid leakage on pulsating flowand heat transfer inside thin films in the presence of flexible sealswere analyzed. Further, as provided herein, flow inside disturbedfluidic cells under wall slip conditions with different fluid types isanalyzed in order to determine their best operating conditions thatcause minimum flow fluctuations. Wall slip conditions can be achievedeither when the fluid contains suspensions or when the substrates arecoated with water repellent resigns. See Watanabe & Udagawa (2001) AIChEJ. 47:256-262, which is herein incorporated by reference. Also, wallslip occurs when the size of the thin film is so small that the Kundsennumber, a ratio of the molecular mean free path to the characteristiclength of the fluidic cell, is between about 10⁻³ to about 10⁻¹ as forflow of gases in microchannels. See Shiping & Ameel (2001) Int. J. Heatand Mass Transfer 44:4225-4234, which is herein incorporated byreference. Thus, as provided herein flow and heat transfer associatedwith side leakage, wall slip condition and non-Newtonian effects insidedisturbed thin films supported by flexible seals are analytically andnumerically examined in order to provide improved fluidic cell designs.

5A. Analysis

A two-dimensional thin film fluidic cell that has a small thickness hcompared to its length 2B and its width D was considered. The inlet ofthis fluidic cell is taken to be at its center forming a symmetricalfluidic cell, as shown in FIG. 43A, in order to assure an almost uniformdeformation in the seal along its length under pulsative flows. Theanalysis was concerned with one half of the fluidic cell shown in FIG.43B due to the symmetry of the proposed cell. The x-axis is taken alongthe axial direction starting from the inlet while y-axis and z-axis aretaken along its thickness and width, respectively, as shown in FIG. 43B.

5B. Fluid Leakage in the Presence of Internal Pressure Pulsations

The lower substrate of the thin film is assumed to be fixed orimmobilized (immobile and inflexible substrate) while the uppersubstrate is attached to the lower substrate by flexible seals andtherefore capable of movement (mobile and inflexible substrate). Theaverage dimensionless motion of the upper substrate H is expressedaccording to the following relation:

$\begin{matrix}{{H \equiv \frac{h}{h_{o}}} = \left( {1 + H_{p}} \right)} & {{Eq}.\mspace{14mu} 106}\end{matrix}$where h, h_(o) and H_(p) are the dimensional average thin filmthickness, a reference thin film thickness and the average dimensionlesschange in the film thickness due to internal pressure forces,respectively.

The following dimensionless variables will be utilized in the analysisherein:

$\begin{matrix}{X = \frac{x}{B}} & {{{Eq}.\mspace{14mu} 107}a} \\{Y = \frac{y}{h_{o}}} & {{{Eq}.\mspace{14mu} 107}b} \\{Z = \frac{z}{B}} & {{{Eq}.\mspace{14mu} 107}c} \\{\tau = {\omega\; t}} & {{{Eq}.\mspace{14mu} 107}d} \\{U = \frac{u}{\left( {{\omega\; B} + V_{o}} \right)}} & {{{Eq}.\mspace{14mu} 107}e} \\{V = \frac{v}{h_{o}\omega}} & {{{Eq}.\mspace{14mu} 107}f} \\{W = \frac{w}{\left( {{\omega\; B} + V_{o}} \right)}} & {{{Eq}.\mspace{14mu} 107}g} \\{\Pi = \frac{p - p_{e}}{{\mu\left( {\omega + \frac{V_{o}}{B}} \right)}ɛ^{- 2}}} & {{{Eq}.\mspace{14mu} 107}h} \\{\theta = \frac{T - T_{1}}{\left( {q_{o}h_{o}} \right)/k}} & {{{Eq}.\mspace{14mu} 107}i}\end{matrix}$where ω, T₁, p_(e), V_(o), μ, k, and ε are the reference frequency ofinternal pulsations, inlet temperature of the fluid, a constantrepresenting the exit pressure, a constant representing a referencedimensional velocity, dynamic viscosity of the fluid, thermalconductivity of the fluid and the perturbation parameter (ε=h_(o)/B),respectively. The pressure at the exit and the outside pressure wereassumed to be at the exit pressure. The lower substrate is maintained ata uniform wall heat flux condition q_(o). The variables t, u, v, w, pand T are the time, axial velocity component, normal velocity component,lateral velocity component, pressure and the temperature, respectively.The dimensionless variables X, Y, Z, τ, U, V, W, Π and θ are thedimensionless forms of x, y, z, t, u, v, w, p and T variables,respectively.

The average dimensionless change in the film thickness was related tothe average dimensionless pressure inside the thin film fluidic cellΠ_(AVG) through the theory of linear elasticity and assumes that thechange in the pressure force on the upper substrate is linearlyproportional to the average change in the thin film thickness by thefollowing relation:H_(p)=F_(n)Π_(AVG)  Eq. 108where F_(n) is named, the fixation parameter. A larger F_(n) valueindicates softer seal-upper substrate assembly. See Boresi et al. (1978)ADVANCED MECHANICS OF MATERIALS Wiley, N.Y., which is hereinincorporated by reference. The inertia of the upper substrate isnegligible because the frequency of pulsations is usually small. Thefixation parameter F_(n) that appears in Equation 108 is equal to:

$\begin{matrix}{F_{n} = {K^{*}\frac{\mu\left( {V_{o} + {\omega\; B}} \right)D}{2\left( {B + {0.5D}} \right)E\; ɛ^{2}h_{s}}}} & {{Eq}.\mspace{14mu} 109}\end{matrix}$where E and h_(s) are the effective modulus of elasticity and theeffective dimension of the seal (h_(s)=h_(o) for a square seal crosssection), respectively. The factor K* reflects the contribution of theelastic behavior of the upper substrate. The parameter F_(n) becomesapparent when the thin film thickness is very small (h_(o) less thanabout 0.15 mm).

Most flows inside thin films possess relatively small Reynolds numbersand could be creep flows as in biological applications, i.e. themodified Reynolds numbers, Re_(L)=V_(o)h_(o)ε/ν and Re_(S)=h_(o) ²ω/ν,are less than one. Therefore, the low Reynolds number flow model wasadopted. Accordingly, the continuity and momentum equations for the flowinside the fluidic cell filled with a Newtonian fluid were reduced tothe following non-dimensionalized equations along with thenon-dimensionalized energy equation:

$\begin{matrix}{U = {\frac{1}{2}\frac{\partial\Pi}{\partial X}{H^{2}\left( \frac{Y}{H} \right)}\left( {\frac{Y}{H} - 1} \right)}} & {{Eq}.\mspace{14mu} 110} \\{V = {\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( {{3\left( \frac{Y}{H} \right)^{2}} - {2\left( \frac{Y}{H} \right)^{3}}} \right)}} & {{Eq}.\mspace{14mu} 111} \\{W = {{- \frac{1}{2}}M_{L}\Pi\;\frac{Z}{H}\left( \frac{Y}{H} \right)\left( {\frac{Y}{H} - 1} \right)}} & {{Eq}.\mspace{14mu} 112} \\{{\frac{\partial^{2}\Pi}{\partial X^{2}} - {\frac{M_{L}}{H^{3}}\Pi}} = {\frac{\sigma}{H^{3}}\frac{\mathbb{d}H}{\mathbb{d}\tau}}} & {{Eq}.\mspace{14mu} 113} \\{{P_{S}\left( {\frac{\partial\theta}{\partial\tau} + {\frac{12}{\sigma}U\frac{\partial\theta}{\partial X}} + {V\frac{\partial\theta}{\partial Y}}} \right)} = \frac{\partial^{2}\theta}{\partial Y^{2}}} & {{Eq}.\mspace{14mu} 114}\end{matrix}$

No slip conditions were assumed at the lower and the upper substrates ofthe fluidic cell as shown in Equation 110. The parameters σ and P_(S)are called the squeezing number and thermal squeezing parameter,respectively, and are defined as:

$\begin{matrix}{\sigma = \frac{12}{1 + \frac{V_{o}}{\omega\; B}}} & {{{Eq}.\mspace{14mu} 115}a} \\{P_{S} = \frac{\rho\; c_{p}h_{o}^{2}\omega}{k}} & {{{Eq}.\mspace{14mu} 115}b}\end{matrix}$

According to Equation 112, the leakage inside the thin film isdistributed equally on both sides of the thin film and it is relativelysmall. Thus, linearization of the lateral pressure gradient was used. Asseen in Equation 112, side leakage is proportional to the pressuredifference between internal and external (at P_(e)) pressures of thethin film. Equation 113 is the corresponding modified Reynolds equationof the problem. Equation 114 is applicable at the plane of symmetry atZ=0. The parameter M_(L) in Equation 112 will be named the dimensionlessleakage parameter and is related to the total leaked mass m_(L) throughthe following relation:

$m_{L} = {\frac{1}{12}{\int_{0}^{1}{M_{L}\ \Pi\;\rho\; D\;{h_{o}\left( {V_{o} + {\omega\; B}} \right)}{{\mathbb{d}X}.}}}}$The inlet pressure due to flow disturbances in the pumping process isconsidered to have the following relation:Π_(i)=Π_(o)(1+β_(v) sin(γωt))  Eq. 116where β_(v), γ, Π_(i) and Π_(o) are the dimensionless amplitude in thepressure, dimensionless frequency of the pressure pulsations, inletdimensionless pressure and the mean dimensionless inlet pressure,respectively. The solution to Equation 113 is obtained as:

$\begin{matrix}{{\Pi\left( {X,\tau} \right)} = {{\left( {\Pi_{i} + {\frac{\sigma}{M_{L}}\frac{\mathbb{d}H}{\mathbb{d}\tau}}} \right){\cosh\left( {\sqrt{\frac{M_{L}}{H^{3}}}X} \right)}} - {\frac{\sigma}{M_{L}}\frac{\mathbb{d}H}{\mathbb{d}\tau}} + {\left( {{\frac{\sigma}{M_{L}}\frac{\mathbb{d}H}{\mathbb{d}\tau}} - {\left\lbrack {\Pi_{i} + {\frac{\sigma}{M_{L}}\frac{\mathbb{d}H}{\mathbb{d}\tau}}} \right\rbrack{\cosh\left( \sqrt{\frac{M_{L}}{H^{3}}} \right)}}} \right)\frac{\sinh\left( {\sqrt{\frac{M_{L}}{H^{3}}}X} \right)}{\sinh\left( \sqrt{\frac{M_{L}}{H^{3}}} \right)}}}} & {{Eq}.\mspace{14mu} 117}\end{matrix}$The reference velocity V_(o) is taken to be the velocity inside the thinfilm in absence of any disturbance and it is related to Π_(o) accordingto following relation:Π_(o)=12−σ  Eq. 1185C. Slip Effects and Non-Newtonian Effects in Presence of ExternalSqueezing

The effects of fluid slip at the boundaries and non-Newtonian effects inthe presence of an external disturbance were analyzed. The dimensionlessoscillation of upper substrate was based on the following genericrelationship:H=1−β cos(γτ)  Eq. 119where β and γ are the amplitude of the motion and a selecteddimensionless frequency, respectively. The apparent viscosity μ of anon-Newtonian fluid such as a biofluid at low flow rates can beexpressed according to the following power-law formula:

$\mu = {\mu_{o}{\frac{\partial u}{\partial y}}^{n - 1}}$where n is a constant representing the power law index. As a result,axial momentum equation for creep flow reduces to the following, μ_(o)replaces μ in Equation 107h:

$\begin{matrix}{\frac{\partial\Pi}{\partial X} = {\left( \frac{V_{o} + {\omega\; B}}{h_{o}} \right)^{n - 1}\frac{\partial}{\partial Y}\left( {{\frac{\partial U}{\partial Y}}^{n - 1}\frac{\partial U}{\partial Y}} \right)}} & {{Eq}.\mspace{14mu} 120}\end{matrix}$

According to the linear relationship between the wall slip velocity andthe shear rate at a solid boundary, the dimensionless boundaryconditions for the axial velocity at the substrates are:

$\begin{matrix}{{{{U\left( {0,\tau} \right)} - {\frac{\beta_{p}}{h_{o}}\frac{\partial{U\left( {0,\tau} \right)}}{\partial Y}}} = 0}{{{U\left( {H,\tau} \right)} + {\frac{\beta_{p}}{h_{o}}\frac{\partial{U\left( {H,\tau} \right)}}{\partial Y}}} = 0}} & {{{{Eqs}.\mspace{14mu} 121}a},{121b}}\end{matrix}$where β_(p) is the dimensional slip parameter. See Navier (1823) Mem.Acad. Sci. Inst. France 1:414-416, which is herein incorporated byreference. By solving Equation 120 and the continuity equation, themodified Reynolds equation is:

$\begin{matrix}{{\frac{\partial}{\partial X}\left\lbrack {\left( {\frac{n}{\left( {{2n} + 1} \right)} + {\left( {2\frac{\beta_{p}}{h_{o}}} \right)\frac{1}{H}}} \right)\left( \frac{H}{2} \right)^{{({{2n} + 1})}/n}\left( {- \frac{\partial\Pi}{\partial X}} \right)^{1/n}} \right\rbrack} = {{- \frac{\sigma}{24}}\frac{\mathbb{d}H}{\mathbb{d}\tau}\left( \frac{V_{o} + {\omega\; B}}{h_{o}} \right)^{{({{2n} + 1})}/n}}} & {{Eq}.\mspace{14mu} 122}\end{matrix}$

For a constant average inlet velocity condition V_(o) during theoscillations, Equation 120 can be used for determining the velocityfield, U and V, for the lower half of the thin film (Y/H<0.5), which arefound to be:

$\begin{matrix}{{U\left( {X,Y,\tau} \right)} = {\frac{\left\lbrack {{{{\sigma\beta\gamma sin}({\gamma\tau})}X} - {\left( {12 - \sigma} \right){H\left( {0,\tau} \right)}}} \right\rbrack}{12\left( {\frac{n}{{2n} + 1} + {2\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}} \right)H} \times {\quad\left\lbrack {{\frac{n}{n + 1}\left\{ {\left( {1 - {2\left( \frac{Y}{H} \right)}} \right)^{(\frac{n + 1}{n})} - 1} \right\}} - {2\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}} \right\rbrack}}} & {{Eq}.\mspace{14mu} 123} \\{{U\left( {X,Y,\tau} \right)} = {\frac{\left\lbrack {{{{\sigma\beta\gamma sin}({\gamma\tau})}X} - {\left( {12 - \sigma} \right){H\left( {0,\tau} \right)}}} \right\rbrack}{12\left( {\frac{n}{{2n} + 1} + {2\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}} \right)H} \times {\quad\left\lbrack {{\frac{n}{n + 1}\left\{ {\left( {1 - {2\left( \frac{Y}{H} \right)}} \right)^{(\frac{n + 1}{n})} - 1} \right\}} - {2\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}} \right\}}}} & {{Eq}.\mspace{14mu} 124}\end{matrix}$

Accordingly, the fluid slip velocity at the wall is obtained as:

$\begin{matrix}{{U_{Slip}\left( {X,\tau} \right)} = {\frac{{- 2}\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}{12\left( {\frac{n}{{2n} + 1} + {2\left( \frac{\beta_{p}}{h_{o}} \right)\frac{1}{H}}} \right)}\mspace{11mu}\frac{\left\lbrack {{{{\sigma\beta\gamma sin}({\gamma\tau})}X} - {\left( {12 - \sigma} \right){H\left( {0,\tau} \right)}}} \right\rbrack}{H}}} & {{Eq}.\mspace{14mu} 125}\end{matrix}$5D. Thermal Boundary Condition

The upper substrate was assumed to be insulated while the lowersubstrate was maintained at a constant heat flux. Accordingly, thedimensionless thermal boundary and initial conditions are:

$\begin{matrix}{{{\theta\left( {X,Y,0} \right)} = 0},{{\theta\left( {0,Y,\tau} \right)} = 0},{\frac{\partial{\theta\left( {X,0,\tau} \right)}}{\partial Y} = {- 1}},{\frac{\partial{\theta\left( {X,H,\tau} \right)}}{\partial Y} = 0}} & {{Eq}.\mspace{14mu} 126}\end{matrix}$5E. Numerical Methods

The dimensionless average thickness of the thin film for the leakageproblem was determined by solving Equations 106 and 108 and the averageof Equation 117, simultaneously, using the explicit formulation withrespect to time. Accordingly, the velocity field U, V and W wasdetermined. The energy equation, Equation 114, was then solved using theAlternative Direction Implicit (ADI) method known in the art bytransferring the problem to one with constant boundaries using thefollowing transformation:

$\begin{matrix}{{\tau^{*} = \tau},{\xi = {{X\mspace{14mu}{and}\mspace{14mu}\eta} = {\frac{Y}{H}.}}}} & \;\end{matrix}$5F. Results and Discussions

The used dimensionless parameters in the leakage problem were selectedaccording to the following data from the literature: the estimatedvolume of the fluidic cell, FIG. 43B, is 50 μl and the flow rate of theliquid is 0.5 ml/min. The thin film thickness was taken to be less thanabout 0.15 mm and the effective modulus of elasticity of the seal wasconsidered to be between 10⁵ pa and 10⁶ pa.

5G. Leakage and Slippage Effects on Flow Dynamics Inside Thin Films

FIG. 44A shows that the thin film thickness decreases as thedimensionless leakage parameter M_(L) increases. A relief in the averageinternal pressure is expected when the leakage rate increases at aconstant inlet pressure. This reduced pressure results in a reduction inthe force holding the upper substrate and a decrease in the thickness.Accordingly, the absolute values of the inlet pressure gradientincreases as the leakage rate increases, thereby causing the inlet flowrate to increase. See FIG. 44B.

According to FIG. 44A, the leakage rate has almost an insignificanteffect on the fluctuation rate at the upper substrate, dH/dτ. However,the associated reduction in the film thickness increases fluctuations inaxial and normal velocities at the sensor position which tend to magnifythe noise in the sensor measurements especially if the sensor is placednear the disturbed substrate. Induced lateral flow due to leakage maycause a lateral bending or twisting of the sensor, e.g. microcantilever.Both effects tend to reduce the accuracy of the measurement and maydamage the microcantilever over a long period of time. The fluctuationsdue to mass leakage can be minimized if the fluidic cell width D ismaximized.

When the upper substrate assembly employs a flexible seal as for largeF_(n) values, the film thickness will be more sensitive to internalpressure pulsations. As a result, an increase in the fixation parameterF_(n) causes an increase in the fluctuation rate at the upper substrateand consequently an increase in flow fluctuations is associated. SeeEquations 110-114 and 117 and FIG. 45. Meanwhile, an increase in thesqueezing number σ results in a reduction in pressure pulsations levels,thereby reducing the fluctuation rate. See FIG. 46. As such, flexibleseals and large velocities produce large fluctuations in the flow withinthe fluidic cell. Similar trends can be extracted for the lateralfluctuations in view of Equation 112. Accordingly, the noise in themeasurement with respect to a microcantilever sensor is magnified as thefixation parameter F_(n) increases especially at large pulsation rates.

The resistance against the flow decreases as the dimensionless wall slipparameter β_(P)/h_(o) increases. Thus, the wall slip velocity increasesas β_(P)/h_(o) increases. See FIG. 47A. This results in a reduction inthe maximum axial velocity since the average flow velocity is keptconstant for each case. The maximum slip velocity occurs during thesqueezing stages. Due to the increase in the uniformity of the axialvelocity profiles as β_(P)/h_(o) increases, flow fluctuations increasenear the fixed substrate (immobile and inflexible substrate). See FIG.47B. This causes enlargement in the noise with respect tomicrocantilever measurements.

Due to the expected increase in wall shear stresses for pseudoplastic(n<1) fluids as the power law index n decreases, the wall slip velocityincreases as n decreases. See FIG. 48A. The uniformity of the axialvelocity profiles increases as n decreases. However, flow fluctuationsincrease near the fixed substrate (immobile and inflexible substrate) asn decreases. See FIG. 48B. Consequently, dilute solutions of testsamples to be analyzed are preferred over undiluted or viscous samplesas they produce minimal flow fluctuations near the undisturbedsubstrate.

5H. Leakage and Slippage Effects on Thermal Characteristics Inside ThinFilms

The reduction in internal pressures associated with an increase in theleakage rate results in an increase in the inlet flow rate which reducesthe average dimensionless lower substrate temperature as seen in FIG.49. This causes the temperature levels around the microcantileversurface to be closer to the inlet temperature. See Equation 107i. Thesetemperatures may be significantly different from the originalmicrocantilever temperature. Thus, the deflection of the bimaterialmicrocantilever due to thermal effects may be magnified when leakage ispresent. Similarly, thermal effects on bimaterial sensors can bemagnified by an increase in F_(n) and a decrease in σ since both effectscause a reduction in the dimensionless average lower substratetemperature. See FIG. 50 and FIG. 51. According to FIG. 49, for therange of M_(L) used, thermal variations can be neglected when comparedto variations in M_(L).

5I. A Design for A Thin Film Fluidic Cell

In order to minimize axial, normal and lateral flow disturbances insidethin films, the parameters F_(n), m_(L) and β are minimized. Designs ofthese films as provided herein can satisfy these constraints. Forexample, the multi-compartment fluidic cell with multiple inlets shownin FIG. 52 will result in reductions in minimizing flow fluctuationsassociated with internal/external disturbances, leakage and thesoftening effect of the support and upper substrate assembly because theexpected reduced pressure difference between the main cell and the twosecondary cells minimizes m_(L). The width of the secondary compartmentis preferably less than half of the width of the main cell, therebyreducing F_(n). The parameters F_(n) and m_(L) can also be reduced ifstiff seals are used for the outer most supports while the interiorflexible seals are kept under compression as shown in FIG. 52. Thus, βis reduced. The flow inside the secondary compartments can be similar tothe primary fluid flow conditions or different.

5J. Conclusions

Flow fluctuations within a fluidic cell and consequently the noise inthe measurement due to flow disturbances, may be minimized byconsidering the following effects:

-   -   Minimizing the working velocities    -   Maximizing the thickness of the upper substrate    -   Maximizing the thin film width if large leakage rate is involved    -   Minimizing the thin film width in the absence of leakage    -   Maximizing the perturbation parameter    -   Utilizing dilute working fluid    -   Maximizing the thin film thickness

The last three effects may increase the microcantilever deflection dueto thermal effects. Thus, in preferred embodiments of the presentinvention, flow oscillations are reduced by employing fluidic celldesigns provided herein.

6. Smart Passive Thermal Systems

6A. Systems with Increased Capacity as Thermal Load Increases

FIG. 53A shows a thin film microchannel with its substrates (inflexible)separated by flexible complex seals, containing closed cavities filledwith a stagnant fluid, such as a gas, possessing a large volumetricthermal expansion. When the thermal load increases over its projectedcapacity, the temperature of the coolant increases causing an increasein the temperature of the thin film substrates. As such the closedcavities get overheated and the stagnant fluid starts to expand. Thiscauses the separation between the thin film substrates to increase whichallows for an increase in the coolant flow rate to increase.Accordingly, the excessive heating is removed. However, under very highoperating thermal conditions, as in lubrications and very high fluxelectrical components, the supporting seals may not work properly.

Thus, the present invention provides an upper substrate (flexible)assembly that can be bent or flexed in certain direction when exposed toheat. Such an upper substrate assembly is shown in FIG. 53B. Inpreferred embodiments, the upper substrate assembly is made to bebimaterial such that the upper layer has a higher linear thermalexpansion coefficient than that for the lower layer material. Excessiveheating causes the coolant temperature to increase which in turn, heatsthe upper substrate assembly. As such, the upper substrate bends outwardallowing for more coolant to flow inside the thin film. In theseembodiments of the present invention, the upper substrate comprises anupper layer and a lower layer, wherein the upper layer comprises amaterial having a linear expansion coefficient that is higher than thatof the material of the lower layer. In some embodiments, the uppersubstrate comprises two metal layers, such as aluminum or gold for theupper layer and copper or bronze for the bottom layer. This can beapplied, for example, when working temperatures exceed about 300° C. andthe thin film thickness is smaller than about 50 μm. In someembodiments, the upper substrate comprises a thermoplastic layer, suchas fluoropolymers including polytetrafluoroethylene,polyperfluoroalkoxyethylene, perfluoromethylvinylether and the like, anda metal layer, such as copper and bronze, and the like. In someembodiments, the upper substrate comprises two thermoplastic layerswherein the linear expansion coefficient of the upper thermoplasticlayer is higher that that of lower thermoplastic layer.

6B. Systems with Decreased Capacity as Thermal Load Increases

FIG. 54A shows a two layered thin film microchannel. The substrates(inflexible) of upper layer which is a secondary fluid layer areseparated by flexible complex seals, containing closed cavities filledwith a stagnant fluid possessing a large volumetric thermal expansion.The lower layer which is the primary fluid layer is only supported byflexible seals. The heated source is connected to the upper substrate.When the thermal load increases over its projected capacity, thetemperature of the secondary fluid passing through the secondary fluidlayer increases causing an increase in the temperature of the uppersubstrate of the two-layered thin film. As such the closed cavities getoverheated and the stagnant fluid starts to expand. This causesshrinkage to the primary fluid layer which reduces the flow rate of theprimary fluid filling the primary fluid layer. This finds itsapplications, among others, in internal combustion where fuel flow isneeded to be reduced as the engine gets overheated.

However, under very high operating thermal conditions, as indeteriorated combustions, the supporting seals may not work properly.Thus, the present invention provides an upper substrate (flexible)assembly that may be operated under high thermal conditions. Such adesign is shown in FIG. 54B. The upper substrate can be made to bebimaterial such that its lower layer has a higher linear thermalexpansion coefficient than that for upper layer material. Excessiveheating causes the coolant temperature to increase which in turn, heatsthe upper substrate. As such, the upper substrate bends inward resultingin less coolant flow rate inside the thin film. In these embodiments ofthe present invention, the upper substrate comprises an upper layer anda lower layer, wherein the upper layer comprises a material having alinear expansion coefficient that is lower than that of the material ofthe lower layer. In some embodiments, the upper substrate comprises twometal layers, such as copper or bronze for the upper layer and aluminumor gold for the lower (This can be applied, for example, when workingtemperatures exceed about 300° C. and the thin film thickness is lowerthan about 50 μm). In some embodiments, the upper substrate comprises anupper metal layer, such as copper or bronze, and a lower thermoplasticlayer such as such as fluoropolymers including polytetrafluoroethylene,polyperfluoroalkoxyethylene, perfluoromethylvinylether, and the like),and a metal layer, such as copper and bronze, and the like. In someembodiments, the upper substrate comprises two thermoplastic layerswherein the linear expansion coefficient of upper thermoplastic layer islower than that that of the lower thermoplastic layer.

EXAMPLE 1 Design of Enhancements in Thermal Characterstics of DifferentInsulating Assemblies Utilizing Expandable Fluid Layers

Generally, thermal losses increase at large working temperatures. Thepresent invention provides an insulating assembly having desirableinsulative attributes at high working temperatures. That is, itseffective thermal resistance increases with an increase in the workingtemperatures. An example of an insulating assembly of the presentinvention is shown in FIG. 1 and is composed of the following frombottom to top: (1) a heated substrate, (2) a layer of fluid that has avery low thermal conductivity such as xenon (the primary fluid layer),(3) a thin layer of an insulating substrate, (4) a secondary fluid layerof another fluid that has a lower thermal conductivity like air (needsto be larger than that of the first layer and is open to the outsideenvironment), and (5) a top insulating substrate. The substrates formingthe fluid layers along with the intermediate insulating substrate wereconnected together by flexible seals. The lower substrate was adjacentor in contact with a heat source. Both the lower substrate and the upperinsulating substrate were fixed (immobile and inflexible substrates)while the intermediate insulating substrate was free to move as it wassupported by flexible seals (mobile and inflexible substrate). In orderto avoid melting of the seals at high temperatures, ordinary homogenousflexible seals can be replaced with closed-cell foams comprising smallair cavities separated by sealed partitions that can sustain hightemperature.

The mathematical modeling for the insulating assembly shown in FIG. 1.The expansion of the primary fluid layer, defined as the change in theprimary fluid layer thickness, Δh₁, divided by the original primaryfluid layer thickness, h_(o), is equal to the following:

$\begin{matrix}{\frac{\Delta\; h_{1}}{h_{o}} = {\left( \frac{T_{o} + {\Delta\; T_{o}}}{2\Delta\; T_{o}} \right)\left\lbrack {\sqrt{\frac{4\left( {T^{*} - T_{o}} \right)\Delta\; T_{o}}{\left( {T_{o} + {\Delta\; T_{o}}} \right)^{2}} + 1} - 1} \right\rbrack}} & {{Eq}.\mspace{14mu} 127}\end{matrix}$where T_(o) and T* are the original primary fluid layer temperature andthe average primary fluid temperature, respectively. The quantity ΔT_(o)is equal to Kh_(o) ²/(m₁R₁) where m₁, R₁ and K are the mass of theprimary fluid, the primary fluid constant and the stiffness of thesupporting seals.

In addition, different plausible insulating assemblies which are compactand can provide additional enhancement to the insulating propertiesutilizing expandable fluid layers. An example of these are shown in FIG.9 where multiple layers of the primary fluid are utilized or balloonsfilled with the primary fluid are used instead of the primary fluidlayers. The enhancement in the insulating properties utilizing differentenhanced insulating assemblies shown in FIG. 1 and FIG. 9 may bedetermined as provided herein using the insulating assemblies shown inFIG. 55A and FIG. 55B. For the setup shown in FIG. 55A, heated air flowsinside a channel with one of its substrates subject to standard fibrousinsulation while the other substrate is subject to the proposedinsulation assembly. In FIG. 55B, both upper and lower substrate of thechannel are subject to the proposed insulation assembly. The arrangementof FIG. 55B is preferred over the arrangement of 55A for hightemperature applications. The heat transfer through the insulatingassembly according to FIG. 55A is:q={dot over (m)} _(air)(c _(p))_(air)(T ₂ −T ₁)  Eq. 128where {dot over (m)}_(air)(c_(p))_(air), T₁ and T₂ are the mass flowrate of the heated air, specific heat of the air, the inlet mean bulktemperature of the air and the exit mean bulk temperature of the air,respectively. This value represents twice the heat transferred througheach insulating assembly in FIG. 55B.

Experiments may be performed under various inlet air temperatures toinvestigate the enhancement in the insulating properties. A sampleresult expected from the experiment is shown in FIG. 56 which shows thatthe insulating assembly shown in FIG. 1 can provide about 10 percentadditional insulating effects at T₁=450 K when flexible seals of thepresent invention are utilized. Useful correlations for the percentsaving in energy are produced when expandable fluid layers are utilized.

EXAMPLE 2 Design of Enhancements in Heat Transfer Inside Expandable ThinFilm Channel Supported by Flexible Complex Seals

FIG. 36 shows a thin film having a flexible complex seal. It is composedof the coolant flow, the working fluid, passage and the sealingassembly. This assembly contains closed cavities filled with a stagnantfluid having a relatively large coefficient of volumetric thermalexpansion. The sealing assembly contains also flexible seals in order toallow the thin film to expand. A candidate for the flexible seal is theclosed cell rubber foam. See Friis et al. (1988) J. Materials Science23:4406-4414, which is herein incorporated by reference. Any excessiveheat transfer to the thin film increases the temperature of thesubstrate. Thus, the stagnant fluid becomes warmer and expands. Theseals are flexible enough so that the expansion results in an increasein the separation between the lower and the upper substrates.Accordingly, the flow resistance of the working fluid passage decreases,causing a flooding of the coolant. As a result, the excessive heatingfrom the source is removed. Flexible seals can be placed between specialguiders as shown in FIG. 1B. As such, side expansion of the seals can beminimized and the transverse thin film thickness expansion is maximized.The prior art provides a theoretical model for flow and heat transferinside an expandable thin film. See Khaled & Vafai (2003) ASME J. HeatTransfer 125:916-925, which is herein incorporated by reference. Theprior art also considers the application of small squeezing Reynoldsnumber. See Khaled & Vafai (2003) ASME J. Heat Transfer 125:916-925,which is herein incorporated by reference. This is present when there isa noise in the thermal load which causes a squeezing effect at the freesubstrate (mobile and inflexible substrate). As such, a model in orderto investigate methods for eliminating the fluctuation rates at the freesubstrate (mobile and inflexible substrate), thereby eliminating flowfluctations in the global system.

The motion of the upper substrate shown in FIG. 36 due to both internalvariations in the stagnant fluid temperature, due to the applied thermalload and the internal pressure is expressed according to the followingrelation:

$\begin{matrix}{{H \equiv \frac{h}{h_{o}}} = \left( {1 + H_{T} + H_{p}} \right)} & {{Eq}.\mspace{14mu} 129}\end{matrix}$where h, h_(o) and H are the thin film thickness, a reference filmthickness and the dimensionless thin film thickness, respectively. Thevariables H_(T) and H_(p) are the dimensionless motion of the uppersubstrate due to the thermal expansion of the stagnant fluid and thedimensionless motion of the upper substrate as a result of thedeformation of seals due to the average internal pressure of the workingfluid, respectively.

The presence of a noise in the thermal load can result in a noise indimensionless film thickness H which produces fluctuations in the flowrate due to squeezing effects. See Khaled & Vafai (2003) ASME J. HeatTransfer 125:916-925, which is herein incorporated by reference. Theflow and heat transfer inside the expandable thin film can be simulatedusing an iterative procedure that results in solving the momentum andenergy equations, Equation 130 and Equation 131, while satisfyingEquation 129.

$\begin{matrix}{{\rho\frac{D\; V}{D\; t}} = {{- {\nabla p}} + {\mu{\nabla^{2}V}}}} & {{Eq}.\mspace{14mu} 130} \\{{\rho\; c_{p}\frac{D\; T}{D\; t}} = {\nabla\left( {k{\nabla T}} \right)}} & {{Eq}.\mspace{14mu} 131}\end{matrix}$where V, T and p are the velocity field vector, temperature and thefluid pressure, respectively, and ρ, μ, c_(p) and k are the primaryfluid's density, primary fluid's absolute viscosity, primary fluid'sspecific heat and the thermal conductivity of the primary fluid,respectively.

For the thin film shown in FIG. 36C, the displacement of the uppersubstrate due to internal pressure variations is related to the averagepressure of the working fluid, P_(AVG), inside the thin film through thetheory of linear elasticity by the following relation:

$\begin{matrix}{H_{p} = {\frac{D\; B}{K\; h_{o}}\left( {p_{AVG} - p_{c}} \right)}} & {{Eq}.\mspace{14mu} 132}\end{matrix}$where D, B, K and p_(e) are the thin film width, thin film length, theeffective stiffness of the sealing assembly and the external pressure,respectively. This is based on the fact that the upper substrate isassumed to be rigid and that the applied force on an elastic material(the flexible seal) is proportional to the elongation of this material.See Norton (1998) MACHINE DESIGN; AN INTEGRATED APPROACH Prentice-Hall,New Jersey, which is herein incorporated by reference.

The increase in the thickness due to a pressure increase in the thinfilm causes a reduction in the stagnant fluid pressure. This actionstiffens the sealing assembly. Therefore, the parameter K is consideredto be the effective stiffness for the sealing assembly and not for theseal itself. When the closed cavities are filled with an ideal gas, theeffective K can be shown to be approximately equal to the following whenthe mass of the stagnant fluid is kept constant for the configurationshown in FIG. 36B:

$\begin{matrix}{K \cong {K_{s\; m}\left\lbrack {1 + \frac{m\; R\; T_{1}}{K_{s\; m}h_{o}^{2}}} \right\rbrack}} & {{Eq}.\mspace{14mu} 133}\end{matrix}$where m, R and K_(sm) are the mass of the ideal fluid in the closedcavities, fluid constant and the stiffness for the pure seal material,respectively.

When check valves are used to ensure that the pressure does not fallbelow the initial stagnant pressure, K is expected to approach K_(sm).Practically, the closed cavity width G is assumed to be large enoughsuch that a small increase in the stagnant fluid pressure due to theexpansion can support the associated increase in the elastic force ofthe seal. Moreover, the fixation parameter can be enhanced by replacingsegments of the seals at different locations by elastic membranesespecially the outermost ones thereby reducing the effective length ofthe seal.

The dimensionless displacement of the upper substrate due to thermalexpansion is related to the difference between the average temperatureof the heated substrate, (T_(W))_(AVG), and the initial stagnant fluidtemperature T₁ by the following linearized model:H _(T) =A*β _(T) C _(F)[(T _(W))_(AVG) −T ₁]  Eq. 134where A* is a constant depending on the closed cavities dimensions andgeometry.

The parameter β_(T) is the volumetric thermal expansion coefficient ofthe stagnant fluid in its approximate form:β_(T)≈(1/V_(s1))(V_(s)−V_(s1))/(T_(s)−T₁)|_(p) _(s1) evaluated at thepressure p_(s1) corresponding to the stagnant fluid pressure at theinlet temperature T₁. The quantities V_(s1) and V_(s) represent theclosed cavity volumes at T₁ and at the present stagnant fluidtemperature T_(s), respectively. The factor C_(F) represents thevolumetric thermal expansion correction factor. This factor isintroduced to account for the increase in the stagnant pressure due toan increase in the elastic force in the seal during the expansion whichtends to decrease the effective volumetric thermal expansioncoefficient. It approaches one as the closed cavity width G increasesand it needs to be determined theoretically. For ideal gases andassembly shown in FIG. 36B, the parameter β_(T) times C_(F) can beapproximated by the following:

$\begin{matrix}{{\beta_{T}C_{F}} \cong \frac{1}{{T_{1}\left( {h_{o}/h_{p_{m}}} \right)} + {\left( {K_{s\; m}h_{o}h_{p_{m}}} \right)/\left( {m\; R} \right)}}} & {{Eq}.\mspace{14mu} 135}\end{matrix}$where h_(pm) is the mean value for the dimensional film thickness priorthermal effects.

A model for evaluating different thermal loads is shown in FIG. 57 andcomprises a thin film supported by flexible complex seal with one inletport and two exit ports. The lower substrate is heated from below usinga heater with a variable capacity. An array of thermocouples areattached beneath the heated substrate which is the lower substrate ofthe thin film and is made from a conductive material. The lowersubstrate temperature is measured at different selected points and thenaveraged for a variety of thermal loads. The upper substrate will betaken as an insulated substrate and the closed cavities are consideredto be insulated in all directions except from the region facing thelower substrate such that the average stagnant fluid temperature isabout equal to the average lower substrate temperature. Moreover, thelower surface of the lower substrate will be considered to be insulated.The experimental results are then compared with model simulations knownin the art. See Khaled & Vafai (2003) ASME J. Heat Transfer 125:916-925,which is herein incorporated by reference.

EXAMPLE 3 Design of the Control of Flow and Thermal Exit ConditionsUsing Two-Layered Thin Films Supported by Flexible Complex Seals

Two-layered thin films possess enhanced cooling capacity. See Vafai &Zhu (1999) Int. J. Heat and Mass Transfer 42:2287-2297, which is hereinincorporated by reference. These two-layered systems also provide apassive control of flow and exit thermal conditions for the main thinfilm when flexible complex seals are separating the substrates of thetwo-layered thin film as shown in FIG. 25. This figure shows that thelower layer contains the primary fluid flow passage where its lowersubstrate is fixed (immobile and inflexible substrate) and its uppersubstrate is free to move in the vertical direction (mobile andinflexible substrate). The flow in the primary fluid layer can be thatof the fuel or fuel-air mixture prior to combustion or flow of abiofluid in a fluidic cell. The upper layer of the thin film contains asecondary fluid flow parallel or counter to the primary fluid flowdirection. The fluid for the secondary fluid layer will be chosen suchthat it will have properties close to the primary fluid flow for fluidiccell applications. This is so that disturbances at the intermediatesubstrate will be diminished. The secondary fluid flow, however, canhave different properties than the primary fluid flow. Such would be thecase when the secondary fluid flow is initiated from external processessuch as combustion residuals or the engine coolant flow.

The upper layer of the two-layered thin film shown in FIG. 25 iscomposed of the secondary fluid flow passage and a sealing assemblywhere its upper substrate is fixed (immobile and inflexible substrate)and subjected to a prescribed heat flux from a heat source. This heatflux can be independent of the primary fluid flow or can be the resultof external processes utilizing the primary fluid flow as in combustionprocesses. The latter can be used for controlling the primary fluid flowconditions while the former may model the increase in the ambienttemperature in a fluidic cell application. This can prevent an increasein the average fluid temperature in an ordinary fluidic cell, therebyavoiding a malfunctioning of the biosensor.

The flexible complex seal of the upper layer contains closed cavitiesfilled with a stagnant fluid having a relatively large volumetricthermal expansion coefficient. This sealing assembly also comprisesflexible seals in order to allow the intermediate substrate to move inthe normal direction. See FIG. 25. Any excessive heating at the uppersubstrate results in an increase in the upper substrate's temperatureresulting in an expansion of the stagnant fluid. This expansion alongwith the increase in inlet pressure in the upper layer, if present,causes the intermediate substrate to move downward. Thus, a compressionin the film thickness of the lower layer is attained resulting in areduction in mass flow rate within the primary fluid flow compartment.This arrangement is utilized to control the combustion rate.

In fluidic cells, excessive heating at the upper substrate causescompression of the primary fluid layer's thickness. Thus, averagevelocity within the primary fluid layer increases, when operated atconstant flow rates, enhancing the convective heat transfer coefficient.This causes the average fluid temperature to approach the lowersubstrate temperature, thereby reducing the bimaterial effects. Whenthis cooling assembly is operated at a constant pressure or at aconstant velocity, the compression of the primary fluid layer due toexcessive heating at the upper substrate reduces the flow rate. Thus,the bulk fluid temperature approaches the lower substrate temperaturewithin a shorter distance. As such, bimaterial effects are also reduced.Flexible seals can be placed between special guiders as shown in FIG.25B. As such, side expansion of the seals can be minimized and thetransverse thin film thickness expansion is maximized.

Both lower and upper substrates were assumed to be fixed (immobile andinflexible substrates) while the intermediate substrate which wasseparated from the lower and upper substrates was free to move only inthe normal direction due to the presence of flexible complex seals(mobile and inflexible substrate). The generic motion of theintermediate substrate due to both variations of the stagnant fluidtemperature in the secondary fluid flow passage and the induced internalpressure pulsations within both the primary fluid and secondary fluidflow passages is expressed according to the following relationship:

$\begin{matrix}{H_{1} = \left( {1 - {A*\beta_{T}{C_{F}\left\lbrack {\left( T_{u} \right)_{AVG} - T_{1o}} \right\rbrack}} + \frac{{D_{1}{B_{1}\left( P_{AVG} \right)}_{1}} - {D_{2}{B_{2}\left( P_{AVG} \right)}_{2}}}{K\; h_{o}}} \right)} & {{Eq}.\mspace{14mu} 136}\end{matrix}$where H₁ (H₁=h₁/h_(o)), (T_(u))_(AVG) and T_(1o) are the dimensionlessdisplacement of the intermediate substrate, average temperature at theupper substrate and the initial stagnant fluid temperature,respectively. The subscript “1” indicates the primary fluid layer while“2” indicates the secondary fluid layer. The flow and heat transferinside the expandable two-layered thin film can be solved using aniterative procedure that results in solving the momentum and energyequations and satisfying Equation 134.

FIG. 58 shows a two-layered thin film supported by flexible complex sealwith two inlet port and four exit ports. The insulating assembly isheated from the top using a heater with a variable capacity. The primaryfluid flow rate is measured experimentally by different methods, such asusing a flowmeter, for different thermal loads. The mean bulktemperature at the exit of the primary fluid layer is also measured fordifferent thermal loads. The experimental results are then compared withthe model simulations presented herein which considers the applicationswhere Reynolds number is small.

EXAMPLE 4 Design of Enhancements in Heat Transfer Inside ExpandableSystems Involving Buoyancy Driven Flows Such as Vertical Channels andOpen Ended Cells Supported by Flexible Complex Seals

Heat transfer and flow induced by either natural or mixed convectioninside vertical channels and open ended cells in the presence offlexible complex seals may be analyzed as provided herein. See FIG. 59Aand FIG. 59B. The insulating assemblies provided herein may be used inelectrical and electronic cooling applications, e.g. placed between twodifferent electronic cards. See Desai et al. (1995) ASME J. ElectronicPackaging 117:34-45, and Daloglu & Ayhan (1999) Int. Communication HeatMass Transfer 6:1175-1182, which are herein incorporated by reference.The heat transfer from these electronic cards can be enhanced if thespacing between these electronic cards is allowed to be expandableaccording to the temperature as when flexible complex seals areutilized. As such, the flexible seals and flexible complex seals of thepresent invention may be used in electronic cooling applications inorder to enhance the operations of the electronic components and toincrease the safety margin for these components.

7. Flexible Microchannel Heat Sink Systems

In this section, single layered (SL) and double layered (DL) flexiblemicrochannel heat sinks are analyzed. The deformation of the supportingseals is related to the average internal pressure by theory ofelasticity. It is found that sufficient cooling can be achieved using SLflexible microchannel heat sinks at lower pressure drop values forsofter seals. Double layered flexible microchannel heat sinks providehigher rate of cooling over SL flexible microchannel heat sinks at thelower range of pressure drops. Single layered flexible microchannel heatsinks are preferred for large pressure drop applications while DLflexible microchannel heat sinks are preferred for applicationsinvolving low pressure drops.

The rapid development of microelectronics has created a need for largeintegration density of chips in digital devices such as VLSI components.These devices require increased current-voltage handling capabilitiesleading to large amount of dissipated heat within a small space.Microchannel heat sinks are one of the proposed methods that can be usedto remove this excessive heating.

Microchannels have a very high heat transfer coefficient. Early works onmicrochannel heat sinks had shown that parallel micro passages with 50μm wide and 302 μm deep had thermal resistances as low as 9×10⁻⁶K/(W/m²). See Tuckerman & Pease (1981) IEEE Electron Device LettEDL-2:126-129. This value is substantially lower than the conventionalchannel sized heat sinks. See Missaggia, L. J., et al. (1989) IEEE J.Quantum Electronics 25:1988-1992; Kleiner, M. B., et al. (1995) IEEETrans on Components, Packaging and Manufacturing Technology Part A18:795-804; and Samalam, V. K. (1989) J. Electronics Materials18:611-617. Microchannel heat sink devices can be used as single layered(SL) micro passage such as those illustrated in the works of Lee andVafai and Fedorov and Viskanta. See Lee & Vafai (1999) Int. J. Heat andMass Transfer 42:1555-1568 and Fedorov & Viskanta (2000) Int. J. Heatand Mass Transfer 43:399-415. Double layered (DL) microchannel heatsinks were introduced for the first time in the work of Vafai and Zhu toprovide additional cooling capacity for the microchannel and to decreasethe axial temperature gradients along the microchannel. See Vafai & Zhu(1999) Int. J. Heat and Mass Transfer 42:2287-2297. Single layeredmicrochannel heat sinks can be either single channel system such asthose analyzed in the work of Harms et. al. or multiple channel system.See Harms, T. M., et al. (1999) Int. J. Heat and Fluid Flow 20:149-157and Lee & Vafai (1999) Int. J. Heat and Mass Transfer 42:1555-1568.

One of the drawbacks of microchannel heat sinks is the increasedtemperature of the coolant as large amount of heat is carried out by arelatively small amount of coolant. As such, new technologies developedin the works of Vafai and Zhu and Khaled and Vafai provides newsolutions for cooling of electronic components utilizing microchannelheat sinks. See Vafai & Zhu (1999) Int. J. Heat and Mass Transfer42:2287-2297; Khaled & Vafai (2002) Int. J. Heat and Mass Transfer45:5107-5115; Khaled & Vafai (2003) ASME J. Heat Transfer 125:916-925;and Khaled & Vafai (2004) Int. J. Heat and Mass Transfer 47:1599-1611.The work of Khaled and Vafai is based on utilizing flexible soft seals.The resulting microchannel heat sink system is referred to as “flexiblemicrochannel heat sink”. See Khaled & Vafai (2002) Int. J. Heat and MassTransfer 45:5107-5115; Khaled & Vafai (2003) ASME J. Heat Transfer125:916-925; and Khaled & Vafai (2004) Int. J. Heat and Mass Transfer47:1599-1611. Khaled and Vafai demonstrated that additional cooling canbe achieved if flexible thin films including flexible microchannel heatsinks are utilized. See Khaled & Vafai (2002) Int. J. Heat and MassTransfer 45:5107-5115. In this work, the expansion of the flexible thinfilm including flexible microchannel heat sink is directly related tothe internal pressure. Khaled and Vafai have demonstrated thatsignificant cooling inside flexible flexible thin films can be achievedif the supporting seals contain closed cavities which are in contactwith the heated surface. See Khaled & Vafai (2003) ASME J. Heat Transfer125:916-925. They referred to this kind of sealing assembly as “flexiblecomplex seals”. Moreover, Khaled and Vafai demonstrated that flexiblecomplex seals along with thin films have important applications indesign and control of the flow and thermal characteristics of thesetypes of systems. See Khaled & Vafai (2004) Int. J. Heat and MassTransfer 47:1599-1611.

In this work, the enhancement in the cooling process inside SL and DLflexible microchannel heat sinks is investigated. The theory of linearelasticity applied to the supporting seals is utilized to relate theaverage internal pressure to the thickness of the flexible microchannelheat sinks. The resulting equations are then solved numerically andanalytically to determine the effects of the pressure drop, softness ofthe supporting seals, the Prandtl number and the coolant mass flow rateon the thermal characteristics of both SL and DL flexible microchannelheat sinks.

The following Table 8 provides the various symbols and meanings used inthis section:

TABLE 8 B microchannel length, m c_(p) specific heat of the coolant, Jkg⁻¹ K⁻¹ W width of the microchannel, m F fixation parameter defined inEq. 145 F_(critical) critical fixation parameter defined in Eq. 155 Hmicrochannel thickness, m H_(o) reference microchannel thickness, mh_(c) convective heat transfer coefficient, W m⁻² K K effectivestiffness of the seal, N m⁻¹ k thermal conductivity of the fluid, W m⁻¹K⁻¹ M dimensionless delivered coolant mass flow rate defined on Eq. 165m dimensional delivered coolant mass flow rate, kg m⁻¹ s⁻¹ Nu lowerplate's Nusselt number defined on Eq. 152 Pr Prandtl number, μc_(p)/k pfluid pressure, N m⁻² q heat flux at the lower plate, W m⁻¹ Re Reynoldsnumber, ρu_(m)H/μ (Re)_(critical) critical Reynolds number defined inEq. 156 Re_(o) dimensionless pressure drop, ρu_(m)H_(o)/μ (Re_(o))_(SL)dimensionless pressure drop for single layered flexible microchannel(Re_(o))_(DL) dimensionless pressure drop for double layered flexiblemicrochannel T, T₁ temperature in fluid and the inlet temperature, K Udimensionless axial velocities, u/u_(m) u dimensional axial velocities,m s⁻¹ u_(m) average axial velocity, m s⁻¹ U_(F) uncertainty in mean bulktemperature with respect to F defined in Eq. 150 U_(Reo) uncertainty inmean bulk temperature with respect to Re_(o) defined in Eq. 149 Xdimensionless axial coordinates, x/H x dimensional axial coordinates, mY dimensionless normal coordinates, y/H y dimensional normalcoordinates, m ε perturbation parameter, H/B ε_(critical) criticalperturbation parameter defined in Eq. 157 ε_(o) reference perturbationparameter, H_(o)/B γ friction force ratio defined in Eq. 165 κ_(m) meanbulk temperature ratio defined in Eq. 161 κ_(w) heated plate temperatureratio defined in Eq. 162 μ dynamic viscosity of the fluid θdimensionless temperature, (T − T₁)/(qH/k) θ_(m) dimensionless mean bulktemperature, (T_(m) − T₁)/(qH/k) θ_(w) dimensionless temperature at theheated plate, (T_(w) − T₁)/(qH/k) θ* temperature normalized withreference conditions defined in Eq. 146 ρ density of the fluid7A. Single Layered Flexible Microchannel Heat Sinks

Consider flow inside a two dimensional microchannel heat sink with aheight H and axial length B. The x-axis is aligned along the channellength while the y-axis is in the traverse direction as shown in FIG.60. The fluid is taken to be Newtonian with constant average properties.Using the following dimensionless variables:

$\begin{matrix}{{X = \frac{x}{B}},{Y = \frac{y}{H}},{U = \frac{u}{u_{m}}},{\theta = \frac{T - T_{1}}{q\;{H/k}}}} & {{Eq}.\mspace{14mu} 137}\end{matrix}$leads to the following dimensionless energy equation:

$\begin{matrix}{{{Re}\;\Pr\; ɛ\; U\frac{\partial\theta}{\partial X}} = \frac{\partial^{2}\theta}{\partial Y^{2}}} & {{Eq}.\mspace{14mu} 138}\end{matrix}$where q, T₁ and Re are the heat flux at the heated plate, the inlettemperature and the Reynolds number (Re=(ρu_(m)H)/μ), respectively. Prand ε are the Prandtl number (Pr=ν/α) and the perturbation parameter(ε=H/B). The mean velocity is related to the pressure drop across thechannel, Δp, through the following relation:

$\begin{matrix}{u_{m} = {\frac{1}{12\mu}\frac{\Delta\; p}{B}H^{2}}} & {{Eq}.\mspace{14mu} 139}\end{matrix}$where μ is the dynamic viscosity of the coolant.

For microchannel heat sinks supported by flexible soft seals, theseparation between the microchannel's plates can be expressed accordingthe following assuming that the seals are linear elastic materials:

$\begin{matrix}{H = {H_{o} + \frac{\Delta\; p\; B\; W}{2K}}} & {{Eq}.\mspace{14mu} 140}\end{matrix}$where H_(o), W and K are a reference thickness of the microchannel heatsink, the width of the microchannel heat sink and the stiffness of thesupporting seal, respectively. As such, the Reynolds number and theperturbation parameter can be expressed according to the followingrelations:Re=Re _(o)(1+Re _(o) F)³  Eq. 141ε=ε_(o)(1+Re _(o) F)  Eq. 142where Re_(o) and ε_(o) are the Reynolds number and the perturbationparameter evaluated at the reference microchannel thickness and theparameter F is the fixation parameter. These parameters are defined as

$\begin{matrix}{{Re}_{o} = {\frac{\rho}{12\mu^{2}}\frac{\Delta\; p}{B}H_{o}^{3}}} & {{Eq}.\mspace{14mu} 143} \\{ɛ_{o} = \frac{H_{o}}{B}} & {{Eq}.\mspace{14mu} 144} \\{F = \frac{6\mu^{2}B^{2}W}{\rho\; K\; H_{o}^{4}}} & {{Eq}.\mspace{14mu} 145}\end{matrix}$

The parameter Re_(o) can be interpreted as the dimensionless pressuredrop parameter. The temperature normalized with respect to the referenceparameters, θ* is defined as follows

$\begin{matrix}{\theta^{*} = \frac{T - T_{1}}{q\;{H_{o}/k}}} & {{Eq}.\mspace{14mu} 146}\end{matrix}$

The normalized mean bulk temperature, obtained from the solution ofintegral form of Eq. 138 is

$\begin{matrix}{\left( \theta^{*} \right)_{m} = \frac{X}{\Pr\;{Re}_{o}{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{3}}} & {{Eq}.\mspace{14mu} 147}\end{matrix}$

The uncertainty in (θ*)_(m), Δ(θ*)_(m), isU _((θ*)) _(m) =Δ(θ*)_(m) =U _(Reo) ΔRe _(o) +U _(H) ΔF   Eq. 148where U_(Reo) and U_(F) are defined as

$\begin{matrix}{U_{{Re}\; o} = {\frac{\partial\left( \theta^{*} \right)_{m}}{\partial{Re}_{o}} = {- \frac{\left( {1 + {4{Re}_{o}F}} \right)X}{\Pr\;{Re}_{o}^{2}{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{4}}}}} & {{Eq}.\mspace{14mu} 149} \\{U_{F} = {\frac{\partial\left( \theta^{*} \right)_{m}}{\partial F} = {- \frac{3X}{\Pr\;{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{4}}}}} & {{Eq}.\mspace{14mu} 150}\end{matrix}$7B. Boundary Conditions

The lower plate is assumed to have a uniform wall heat flux and theupper plate is considered to be insulated. As such the dimensionlessboundary conditions can be written as

$\begin{matrix}{{{\theta\left( {0,Y} \right)} = 0},{\frac{\partial{\theta\left( {X,0} \right)}}{\partial Y} = {- 1}},{\frac{\partial{\theta\left( {X,1} \right)}}{\partial Y} = 0}} & {{Eq}.\mspace{14mu} 151}\end{matrix}$

The Nusselt number is defined as

$\begin{matrix}{{Nu} = {\frac{h_{c}H_{o}}{k} = {\frac{1}{\left( \theta^{*} \right)_{W} - \left( \theta^{*} \right)_{m}} = \frac{1}{{\theta^{*}\left( {X,0} \right)} - \left( \theta^{*} \right)_{m}}}}} & {{Eq}.\mspace{14mu} 152}\end{matrix}$where (θ*)_(W) is the heated plate temperature normalized with respectto the reference parameters. Under fully developed thermal conditions,Nusselt number approaches the following value:

$\begin{matrix}{{Nu} = {\frac{h_{c}H_{o}}{k} = {\frac{2.69}{1 + {{Re}_{o}F}} = \frac{1}{\left( \theta^{*} \right)_{W} - \left( \theta^{*} \right)_{m}}}}} & {{Eq}.\mspace{14mu} 153}\end{matrix}$where (θ*)_(W) is the dimensionless lower plate temperature under fullydeveloped thermal conditions. Thus, it can be expressed according to thefollowing:

$\begin{matrix}{\left\lbrack \left( \theta^{*} \right)_{W} \right\rbrack_{{f\; d}\;} = {\frac{1 + {{Re}_{o}F}}{2.69} + \frac{X}{\Pr\;{Re}_{o}{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{3}}}} & {{Eq}.\mspace{14mu} 154}\end{matrix}$

Minimizing this temperature at the exit results in the following valueof the fixation parameter

$\begin{matrix}{\frac{\partial\left\lbrack \left( \theta^{*} \right)_{W} \right\rbrack_{{f\; d}\;}}{\partial F} = {\left. 0\Rightarrow F_{critical} \right. = {\frac{1.685}{\left( {{Re}_{o}\Pr\; ɛ_{o}} \right)^{1/4}{Re}_{o}} - \frac{1}{{Re}_{o}}}}} & {{Eq}.\mspace{14mu} 155}\end{matrix}$

As such, the corresponding Reynolds number and the perturbationparameters are

$\begin{matrix}{({Re})_{critical} = {4.784\left( \frac{{Re}_{o}}{\left( {\Pr\; ɛ_{o}} \right)^{3}} \right)^{1/4}}} & {{Eq}.\mspace{14mu} 156} \\{\left( ɛ_{o} \right)_{critical} = {1.685\left( \frac{ɛ_{o}^{3}}{{Re}_{o}\Pr} \right)^{1/4}}} & {{Eq}.\mspace{14mu} 157}\end{matrix}$7C. Double Layered Flexible Microchannel Heat Sinks

FIG. 61 shows the proposed two layered (DL) flexible microchannel heatsink with counter flow as proposed by Vafai and Zhu. See Vafai & Zhu(1999) Int. J. Heat and Mass Transfer 42:2287-229. The governing energyequations for both layers are

$\begin{matrix}{{{Re}_{o}\Pr\;{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{4}{U\left( Y_{1} \right)}\frac{\partial\theta_{1}}{\partial X_{1}}} = \frac{\partial^{2}\theta_{1}}{\partial Y_{1}^{2}}} & {{Eq}.\mspace{14mu} 158} \\{{{Re}_{o}\Pr\;{ɛ_{o}\left( {1 + {{Re}_{o}F}} \right)}^{4}{U\left( Y_{2} \right)}\frac{\partial\theta_{2}}{\partial X_{2}}} = \frac{\partial^{2}\theta_{2}}{\partial Y_{2}^{2}}} & {{Eq}.\mspace{14mu} 159}\end{matrix}$where the subscripts 1 and 2 are for the lower and the upper layers,respectively. The corresponding boundary conditions are

$\begin{matrix}{{{{\theta_{1}\left( {{X_{1} = 0},Y} \right)} = {{\theta_{2}\left( {{X_{2} = 0},Y} \right)} = 0}},{\frac{\partial{\theta_{1}\left( {X_{1},0} \right)}}{\partial Y_{1}} = {- 1}}}{{\frac{\partial{\theta_{1}\left( {X_{1},1} \right)}}{\partial Y_{1}} = \frac{\partial{\theta_{2}\left( {{X_{2} = {1 - X_{1}}},0} \right)}}{\partial Y_{2}}},{\frac{\partial{\theta_{2}\left( {X_{2},1} \right)}}{\partial Y_{2}} = 0}}} & {{Eq}.\mspace{14mu} 160}\end{matrix}$

The intermediate plate is taken to be made from a highly conductivematerial like copper such that temperature variation across this plateis negligible. The following parameters are introduced in order tocompare the performance of the DL flexible microchannel heat sinkcompared to SL flexible microchannel heat sink:

$\begin{matrix}\begin{matrix}{{\kappa_{m} = \frac{\left\lbrack {\theta_{m\; 1}^{*}\left( {X_{1} = 1} \right)} \right\rbrack_{D\; L}}{\left\lbrack {\theta_{m}^{*}\left( {X = 1} \right)} \right\rbrack_{S\; L}}},} \\{\kappa_{W} = \frac{\left\lbrack \left( \theta_{W}^{*} \right)_{AVG} \right\rbrack_{D\; L}}{\left\lbrack \left( \theta_{W}^{*} \right)_{AVG} \right\rbrack_{S\; L}}}\end{matrix} & {{{Eqs}.\mspace{14mu} 161},162}\end{matrix}$Lower values of the cooling factors κ_(m) and κ_(W) indicate that DLflexible microchannel heat sinks are preferable over SL flexiblemicrochannel heat sinks.

Another factor that will be considered is the ratio of the totalfriction force in DL flexible microchannel heat sinks to that for SLflexible microchannel heat sinks delivering the same flow rate ofcoolant. It can be shown that this factor is equal to

$\begin{matrix}\begin{matrix}{\gamma \equiv \frac{\left( {{Friction}\mspace{14mu}{force}} \right)_{D\; L}}{\left( {{Friction}\mspace{14mu}{force}} \right)_{S\; L}}} \\{= \frac{2\left( {\Delta\; p} \right)_{D\; L}H_{D\; L}}{\left( {\Delta\; p} \right)_{S\; L}H_{S\; L}}} \\{= \frac{2\left( {Re}_{o} \right)_{D\; L}\left( {1 + {\left( {Re}_{o} \right)_{D\; L}F}} \right)}{\left( {Re}_{o} \right)_{S\; L}\left( {1 + {\left( {Re}_{o} \right)_{S\; L}F}} \right)}}\end{matrix} & {{Eq}.\mspace{14mu} 163}\end{matrix}$where (Re_(o))_(DL) and (Re_(o))_(SL) are related through the following:(Re _(o))_(DL)(1+(Re _(o))_(DL) F)³=2(Re _(o))_(SL)(1+(Re _(o))_(SL)F)³  Eq. 164

As such, the delivered dimensionless mass flow rate by both SL and DLflexible microchannel heat sinks is

$\begin{matrix}{M = {\frac{m}{\mu} = {\frac{\left( {2\rho\; u_{m}H} \right)}{\mu} = {2\left( {Re}_{o} \right)_{D\; L}\left( {1 + {\left( {Re}_{o} \right)_{D\; L}F}} \right)^{3}}}}} & {{Eq}.\mspace{14mu} 165}\end{matrix}$where m is the dimensional mass delivered by both flexible microchannelheat sinks.7D. Numerical Analysis

Equations 137, 158 and 159 were descritized using three points centraldifferencing in the transverse direction while backward differencing wasutilized for the temperature gradient in the axial direction. Theresulting tri-diagonal system of algebraic equations at X=ΔX was thensolved using the well established Thomas algorithm. See Blottner, F. G.(1970) AIAA J. 8:193-205. The same procedure was repeated for theconsecutive X-values until X reached the value of unity. For equations158 and 159, the temperature distribution at the intermediate plate wasinitially prescribed. Equations 158 and 159 were solved as describedbefore. The thermal boundary condition at the intermediate plate wasthen used to correct for intermediate plate temperatures. The procedurewas repeated until all the thermal boundary conditions were satisfied.

In most of the cases considered here, the minimum value of Re was takento be 50 while the maximum Re value was allowed to expand to 2100 forRe_(o)=50 and F=0.05. The maximum Re corresponded to a microchannel heatsink that was substantially expanded due to the presence of soft seals.The thickness for the latter limiting case (Re=2100) was found to be 3.5times the thickness of the former limiting case (Re=50). The maximumfixation parameter was taken to be 0.05. This represented a thin filmmicrochannel heat sink filled with water, having B=60 mm, W=20 mmh_(o)=0.3 mm, and K=1000 N/m.

7E. Results

7E1. Effects of Fixation Parameter and Pressure Drop on the ThermalBehavior of SL Flexible Microchannel Heat Sinks

FIGS. 62 and 63 illustrate effects of the fixation parameter F and thedimensionless pressure drop Re_(o) on the mean bulk temperature at theexit and the average heated plate temperature for SL flexiblemicrochannel heat sinks, respectively. As the seal become softer, thefixation parameter increases allowing for further expansion of themicrochannel at a given dimensionless pressure drop, Re_(o). Thus, themean bulk temperature is further reduced as shown in FIG. 62 and theheated plate is further cooled as shown in FIG. 63 due to an increase inthe coolant flow rate. As seen in FIG. 63, relatively low pressure dropis capable of producing efficient cooling compared to that at largerpressure drops for larger F values.

Convective heat transfer coefficient is reduced as F increases at lowdimensionless pressure drops as shown in FIG. 64. This is becausecoolant velocities decrease near the heated plate as F increases.However, for larger pressure drops, flow increases due to both anincrease in the pressure drop and the expansion of the microchannel as Fincreases resulting in an increase in the thermal developing regioneffects. As such, the convective heat transfer coefficient increases asF increases for larger Re_(o) and F values as illustrated in FIG. 64.FIG. 65 shows that the mean bulk temperature becomes less sensitive tothe dimensionless pressure drop Re_(o) and the fixation parameter F asboth F and Re_(o) increase.

FIG. 66 demonstrates that flexible microchannel heat sinks operating atlower Reynolds numbers possess lower heated plate temperature at theexit as F increases. This is not seen when these heat sinks are operatedat higher Reynolds number values. As such, the enhancement in thecooling process using flexible microchannel heat sinks is notsignificant at large pressure drops as illustrated in FIG. 63.

7E2. Effects of Fixation Parameter and Prandtl Number on ThermalBehavior of SL Flexible Microchannel Heat Sinks.

FIG. 67 illustrates the effects of the fixation parameter F and Prandtlnumber Pr on the average heated plate temperature for SL flexiblemicrochannel heat sinks. As seen in FIG. 67, sufficient increase in thecooling effect can be achieved by increasing F as Pr decreases. This ismainly due to an increase in the coolant flow rate as F increases. Onthe other hand, convective heat transfer coefficient is reduced as Fincreases at low Pr values as shown in FIG. 68. This is because coolantvelocities decrease near the heated plate as F increases. As seen inFIG. 68 for large Pr values, thermal developing region effects increasecausing the convective heat transfer coefficient to increase as Fincreases.

7E3. Effects of Fixation Parameter and Pressure Drop on Thermal Behaviorof DL Flexible Microchannel Heat Sinks.

FIG. 69 describes the axial behavior of the mean bulk temperature fortwo different DL flexible microchannel heat sinks having differentfixation parameters. Additional cooling is achieved by introducing thesecondary layer which can be seen in FIG. 69 for the case with F=0.01.This plot shows that the maximum coolant temperature occurs before theexit unlike SL flexible microchannel heat sinks where this temperatureoccurs at the exit. As F increases, convection increases in the mainlayer while conduction to the upper layer decreases. This is due to anincrease in the convective heat transfer and an increase in theexpansion of the main layer. As such, the increase in the coolingcapacity of DL flexible microchannel heat sinks becomes insignificant atboth large values of the pressure drop and the fixation parameter. Thisfact is clearly seen in FIG. 70 where the heated plate temperature forDL flexible microchannel heat sinks are almost the same as that for theSL flexible microchannel heat sinks with F=0.05 for a wide range ofRe_(o). Note that κ_(m) is the ratio of the mean bulk temperature at theexit for DL flexible microchannel to that for SL flexible microchannelheat sink. The parameter κ_(W) is the ratio of the average heated platetemperature for DL flexible microchannel to that for SL flexiblemicrochannel heat sink.

7E4. Comparisons between SL and DL Flexible Microchannel Heat SinksDelivering the Same Coolant Flow Rates.

FIG. 71 shows the effect of the fixation parameter F and thedimensionless pressure drop for DL flexible microchannel heat sinks onthe pressure drop and friction force ratios between SL and DL flexiblemicrochannel heat sinks. These microchannel heat sinks are considered todeliver the same coolant flow rate. As F increases, the pressure drop inSL flexible microchannel heat sinks required to deliver the same flowrate as for the DL flexible microchannel heat sinks decreases. Thisvalue is further decreased as the pressure drop in DL flexiblemicrochannel heat sink increases. Meanwhile, as F increases, the ratioof the friction force encountered in the proposed DL flexiblemicrochannel heat sink to that associated with the SL flexiblemicrochannel heat sink increases. This indicates that SL flexiblemicrochannel heat sinks delivering the same flow rate as for DLmicrochannel heat sinks having the same F value encounter fewer frictionlosses.

FIG. 72 demonstrates that SL flexible microchannel heat sinks canprovide better cooling attributes compared to DL flexible microchannelheat sinks delivering the same coolant flow rate and having the same Fvalues. However; note that rigid DL microchannel heat sinks providesbetter cooling than rigid SL microchannel heat sinks when operated atthe same pressure drop as shown in FIG. 70. It should be noted that FIG.72 shows that microchannel heat sinks with stiffer seals provideadditional cooling over those with softer seals delivering the same flowrate. This is because the former are thinner and have larger velocitiesthan the latter microchannel heat sinks. As such, convective heattransfer for rigid microchannels will be higher than that for flexiblemicrochannel heat sinks delivering the same flow rate.

7F. Conclusions

Heat transfer inside SL and DL flexible microchannel heat sinks havebeen analyzed in this work. The deformation of the supporting seals wasrelated to the average internal pressure by theory of linear elasticity.Increases in the fixation parameter and the dimensionless pressure dropwere found to cause enhancements in the cooling process. Theseenhancements are significant at lower pressure drop values. Moreover, DLflexible microchannel heat sinks were found to provide additionalcooling which were significant at lower values of pressure drop forstiff seals. It is preferred to utilize SL flexible microchannel sinksover DL microchannel heat sinks for large pressure drop applications.However, at lower flow rates the DL flexible microchannel heat sink ispreferred to be used over SL flexible microchannel heat sinks especiallywhen stiff sealing material is utilized.

8. Heat Transfer Enhancement Through Control of Thermal DispersionEffects

Heat transfer enhancements are investigated inside channels bycontrolling thermal dispersion effects inside the fluid. Differentdistributions for the dispersive elements such as nanoparticles orflexible hairy fins extending from the channel plates are considered.Energy equations for different fluid regions are dimensionalized andsolved analytically and numerically. The boundary arrangement and theexponential distribution for the dispersive elements are found toproduce enhancements in heat transfer compared to the case with auniform distribution for the dispersive elements. The presence of thedispersive elements in the core region does not affect the heat transferrate. Moreover, the maximum Nusselt number for analyzed distributions ofthe dispersive elements are found to be 21% higher than that withuniformly distributed dispersive elements for a uniform flow. On theother hand, the parabolic velocity profile is found to produce a maximumNusselt number that is 12% higher than that with uniformly distributeddispersive elements for the boundary arrangement. The distribution ofthe dispersive elements that maximizes the heat transfer is governed bythe flow and thermal conditions plus the properties of the dispersiveelements. Results in this work point towards preparation of supernanofluids or super dispersive media with enhanced coolingcharacteristics.

In some embodiments, the super dispersive media comprises at least onenanoparticle which may be metallic or carbon based and include nanotubesand flexible nanostrings known in the art. In preferred embodiments, thedevices of the present invention comprise a coolant and super dispersivemedia in the microchannels, preferably the super dispersive mediacomprises at least one metallic nanoparticle, at least one carbonnanoparticle, at least one nanotube, at least one flexible nanostring,or a combination thereof.

In some embodiments, the super dispersive media is non-uniformlydistributed in the volumetric space of the microchannel. In someembodiments, the super dispersive media is minimally distributed in thevolumetric space of microchannel regions having least transverseconvection heat transfer. In other words, the concentration of the superdispersive media is minimal in the volumetric space of microchannelregions having least transverse convection heat transfer. In someembodiments, the super dispersive media is maximum in the volumetricspace of microchannel regions having maximum transverse convection heattransfer.

The following Table 9 provides the various symbols and meanings used inthis section:

TABLE 9 B channel length C* dispersive coefficient (dependent on thedispersive elements properties) c_(p) average specific heat E_(o)thermal dispersion parameter h Half channel height h_(c) convective heattransfer coefficient k thermal conductivity k_(o) effective staticthermal conductivity of the nanofluid Nu Nusselt number Nu_(fd) Nusseltnumber at fully developed condition P_(e) Peclet number q heat flux atthe channel walls T, T₁ fluid's temperature and the inlet temperature U,u dimensionless and dimensional axial velocities X, x dimensionless anddimensional axial coordinates Y, y dimensionless and dimensional normalcoordinates θ, θ_(m) dimensionless temperature and dimensionless meanbulk temperature θ_(w) dimensionless temperature of the channel plates ρdensity f pure fluid nf nanofluid p particle

The heat flux of VLSI microelectronic components can reach up to 1000kW/m². As such, many methods are proposed to eliminate excess of heatingassociated with the operation of these components. One of these methodsis to utilize two-layered microchannels. See Vafai & Zhu (1999) Int. J.Heat Mass Transfer 42:2287-2297. Two phase flow are utilized for coolingwhich was found to be capable of removing maximum heat fluxes generatedby electronic packages yet the system may become unstable near certainoperating conditions. See Bowers & Mudawar (1994) ASME J. ElectronicPackaging 116:290-305. The use of porous blocks inside channels wasfound to be efficient in eliminating the excess of heat. See Vafai &Huang (1994) ASME J. Heat Transfer 116:604-613; Huang & Vafai (1994)AIAA J. Thermophysics and Heat Transfer 8:563-573; and Hadim, A. (1994)ASME J. Heat Transfer 116:465-472. However, the porous medium creates asubstantial increase in the pressure drop inside the cooling device.Recently, Khaled and Vafai demonstrated that expandable systems canprovide an efficient method for enhancing the cooling rate. See Khaled &Vafai (2003) ASME J. Heat Transfer 125:916-925. The performance ofexpandable systems and other cooling systems can be further improvedwhen nanofluids are used as their coolants. See Khaled & Vafai (2003)ASME J. Heat Transfer 125:916-925; Khaled & Vafai (2002) Numerical HeatTransfer, Part A, 42:549-564; Khanafer, K., et al. (2003) Int. J. HeatMass Transfer 46:3639-3653; and Vafai & Khaled (2004) Int. J. Heat MassTransfer 47:743-755.

Nanofluids are mixtures of a pure fluid with a small volume ofsuspensions of ultrafine particles such as copper nanoparticles ornanotubes. They were found to possess a large effective thermalconductivity. For example, the effective thermal conductivity ofnanofluids could reach 1.5 times that of the pure fluid when the volumefraction of the copper nanoparticles is 0.003. See Eastman, J. A., etal. (2001) Applied Physics Letters 78:718-720. This enhancement isexpected to be further enhanced as the flow speed increases resulting inan increase in the mixing effects associated with the Brownian motion ofthe nanoparticles. This mixing effect is referred in literature as thethermal dispersion effect. See Xuan & Roetzel (2000) Int. J. Heat MassTransfer 43:3701-3707. Other aspects of dispersion effects can be foundin some of the recent works. See Chang, P. Y., et al. (2004) NumericalHeat Transfer, 45:791-809; Hancu, S., et al. (2002) Int. J. Heat MassTransfer 45:2707-2718; Kuznetsov, A. V., et al. (2002) Numerical HeatTransfer 42:365-383; Gunn, D. J. (2004) Int. J. Heat Mass Transfer48:2861-2875; and Metzger, T., et al. (2004) Int. J. Heat Mass Transfer47:3341-3353. Li and Xuan (Li & Xuan (2002) Science in China (Series E)45:408-416) reported an increase of 60% in the convective heat transferinside a channel filled with a nanofluid, having 3% volume fraction forcopper nanoparticles, compared to its operation with the pure fluid.This significant increase indicates that thermal dispersion is the mainmechanism for heat transfer inside convective flows. The challenge is tofind new ways to improve the performance of the cooling systems.

In this work, a method for enhancing the heat transfer characteristicsthrough the use of nanofluids with proper thermal dispersion propertiesis proposed and analyzed. This can be accomplished by having a properdistribution for the ultrafine particles. Physically, the distributionof the ultrafine particles can be controlled using different methods:(i) having nanoparticles with different sizes or physical properties,(ii) applying appropriate magnetic forces along with using magnetizednanoparticles, (iii) applying appropriate centrifugal forces, and (iv)applying appropriate electrostatic forces along with using electricallycharged nanoparticles. Different distribution for the nanoparticles canbe obtained using any combination of the above methods.

For example, denser nanoparticles such as copper nanoparticles or thosewith a larger size tend to suspend at lower altitudes in coolants.However, nanoparticles with lower density such as carbon nanoparticlesor those having a lower size tend to swim at higher altitudes withindenser liquids such as aqueous solutions and liquid metals. As such,non-homogenous thermal dispersion properties can be attained.Centrifugal effects tend to produce concentrated thermal dispersionproperties near at least one of the boundaries. On the other hand,non-homogenous thermal dispersion properties inside the coolant can beobtained by attaching to the plates of the cooling device flexible thinfins like hair with appropriate lengths. The Brownian motion of thesuspended hairy medium will increase the thermal dispersion propertiesmainly near the plates of the cooling device and it can be used with aproper suspension system to obtain any required thermal dispersionproperties.

Heat transfer enhancements are analyzed inside a channel filled with acoolant having different thermal dispersion properties. Differentarrangements for the nanoparticles or the dispersive elements areconsidered in this work. The nanoparticles or the dispersive elementsare considered to be uniformly distributed near the center of thechannel for one of the arrangements. In another arrangement, they areuniformly distributed near the channel plates. Exponential or parabolicdistributions for the dispersive elements are also analyzed in thiswork. The energy equations for the corresponding fluid regions arenon-dimensionalized. Solutions for the Nusselt number and thetemperature are obtained analytically for special cases and numericallyfor general cases. They are utilized to determine the appropriatedistribution for the dispersive elements that will result in the maximumheat transfer with the same total number of nanoparticles or thedispersive elements.

8A. Problem Formulation

Consider a flow inside a two dimensional channel with a height 2 h and alength B. The x-axis is aligned along the centerline of the channelwhile the y-axis is in the traverse direction as shown in FIG. 73. Thefluid which could be a pure fluid or a nanofluid is taken to beNewtonian with constant average properties except for the thermalconductivity to account for thermal dispersion effects. The energyequation is:

$\begin{matrix}{{\rho\; c_{p}u\frac{\partial T}{\partial x}} = {\frac{\partial}{\partial y}\left( {k\frac{\partial T}{\partial y}} \right)}} & {{Eq}.\mspace{14mu} 166}\end{matrix}$where T, ρ, c_(p) and k are the temperature, effective fluid density,fluid specific heat and thermal conductivity, respectively. The velocityfield u in the channel is taken to be fully developed. The volume of thedispersive elements is very small such that the velocity profile isparabolic.

$\begin{matrix}{\frac{u}{u_{m}} = {\frac{3}{2}\left( {1 - \left( \frac{y}{h} \right)^{2}} \right)}} & {{Eq}.\mspace{14mu} 167}\end{matrix}$where u_(m) is the mean flow speed.

For nanofluids or in the thermally dispersed region, the parameterρc_(p) will be (ρc_(p))_(nf) and it is equal to(ρc _(p))_(nf)=(1−φ)(ρc _(p))_(f)+φ(ρc _(p))_(p)  Eq. 168where the subscript nf, f and p denote the nanofluid or the dispersiveregion, pure fluid and the particles, respectively. The parameter φ isthe nanoparticles or the dispersive elements volume fraction whichrepresents the ratio of the nanoparticles or the dispersive elementsvolume to the total volume. A nanofluid composed of pure water andcopper nanoparticles suspensions with 2% volume fraction has a value of(ρc_(p))_(nf) equal to 99% that for the pure water which is almost thesame as the thermal capacity of the pure fluid.

The ultrafine suspensions such as nanoparticles, nanotubes or anydispersive elements in the fluid plays an important role in heattransfer inside the channel as their Brownian motions tend to increasefluid mixing. This enhances the heat transfer. The correlationspresented in the work of Li and Xuan (Li & Xuan (2002) Science in China(Series E) 45:408-416) for Nusselt numbers in laminar or turbulent flowsshow that the heat transfer is enhanced in the presence of nanoparticlesand it increases as the nanoparticles volume fraction, the diameter ofthe nanoparticles or the flow speed increase. Xuan and Roetzel (Xuan &Roetzel (2000) Int. J. Heat Mass Transfer 43:3701-3707) suggest(consistent with the dispersion model given in Amiri and Vafai (Amiri &Vafai (1994) Int. J. Heat Mass Transfer 37:939-954)) the followinglinearalized model for the thermal conductivity of the nanofluid:k=k _(o) +C*(ρc _(p))_(nf) φhu  Eq. 169where C* is a constant depending on the diameter of the nanoparticle andits surface geometry.

Physically, Equation 169 is a first approximation for the thermalconductivity of the nanofluid that linearly relates it to thermalcapacitance of the flowing nanoparticles or flowing dispersive elements.The constant k_(o) represents the effective thermal conductivity of thenanofluid or the dispersive region under stagnant conditions, at u=0.This constant can be predicted for nanofluids from the formula suggestedby Wasp (Wasp, F. J. (1977) Solid-Liquid Slurry Pipeline Transportation,Trans. Tech. Berlin) which has the following form:

$\begin{matrix}{\frac{k_{o}}{k_{f}} = \frac{k_{p} + {2k_{f}} - {2{\varphi\left( {k_{f} - k_{p}} \right)}}}{k_{p} + {2k_{f}} + {2{\varphi\left( {k_{f} - k_{p}} \right)}}}} & {{Eq}.\mspace{14mu} 170}\end{matrix}$where k_(p) and k_(f) are the thermal conductivity of the nanoparticlesand the pure fluid, respectively.

According to Equation 170, a two percent volume fraction of ultrafinecopper particles produces 8 percent increase in k_(o) when compared tothe thermal conductivity of the pure fluid. On the other hand, theexperimental results illustrated in the work of Li and Xuan (Li & Xuan(2002) Science in China (Series E) 45:408-416) shows that the presenceof suspended copper nanoparticles with 2 percent volume fractionproduced about 60% increase in the convective heat transfer coefficientcompared to pure fluid.

See Table 10 as follows:

TABLE 10 Variations of (ρc_(p))_(nf)/(ρc_(p))_(f) and k_(o)/k_(f) forvarious ultrafine copper particles volume ratios φ (percent)(ρc_(p))_(nf)/(ρc_(p))_(f) k_(o)/k_(f) 0 1 1 1 0.998 1.040 2 0.996 1.0833 0.995 1.127 4 0.993 1.173 5 0.991 1.221

This indicates that thermal dispersion is the main mechanism forenhancing heat transfer inside channels filled with nanofluids underconvective conditions. Non-dimensionalizing Equation 166 with thefollowing dimensionless variables:

$\begin{matrix}{{X = \frac{x}{h}},{Y = \frac{y}{h}},{U = \frac{u}{u_{m}}},{\theta = \frac{T - T_{1}}{q\;{h/k_{f}}}}} & {{Eq}.\mspace{14mu} 171}\end{matrix}$leads to the following dimensionless energy equation:

$\begin{matrix}{{P_{e}U\frac{\partial\theta}{\partial X}} = {\frac{\partial}{\partial Y}\left( {\frac{k}{k_{f}}\frac{\partial\theta}{\partial Y}} \right)}} & {{Eq}.\mspace{14mu} 172}\end{matrix}$where q, T₁ and Pe are the heat flux at the channel's plates, the inlettemperature and the Peclet number (P_(e)=(ρc_(p)u_(m)h)/k_(f)),respectively. It is assumed that the heat flux is constant and equal atboth plates.

For simplicity, the term k/k_(f) will be rearranged in the followingform:

$\begin{matrix}{{\frac{k}{k_{f}} = {\frac{k_{o}}{k_{f}} + {\lambda\frac{\;\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\; U_{n\; f}}}},{\lambda = {C*{Pe}_{f}}}} & {{Eq}.\mspace{14mu} 173}\end{matrix}$where Pe_(f)=(ρc_(p))_(f)u_(m)h/k_(f).

A portion of the fluid's volume are considered in part of this work tobe subjected to thermal dispersion effects due the suspensions ofnanoparticles or any dispersive elements while the other portioncontains only the pure fluid. The most obvious way to obtain specificdistributions for thermal dispersive elements is to have conductivehairy fins extending from the channel plates or from a carefullydesigned fixed or flexible structure placed in the channel. The volumeof this structure is small enough such that the parabolic assumption forthe velocity profile is still valid. Also, non-homogenous thermaldispersion properties can be achieved by having nanoparticles withdifferent densities or different sizes. Heavier nanoparticles ordispersive elements tend to swim closer to the lower plate due togravitational forces while lighter nanoparticles or dispersive elementstend to swim closer to the upper force due to buoyancy forces. Thedispersive elements such as nanoparticles can be further concentratednear the channel's plates by having these particles magnetized alongwith applying appropriate magnetic fields. As such, the difference inthe thermal dispersive properties of the nanofluid can be achieved.Appropriate thermal dispersive properties can be obtained by utilizingthe different methods discussed in the introduction section.

The dimensionless energy equation for the part involving thermaldispersion is

$\begin{matrix}{{\left( P_{e} \right)_{f}\left( \frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}} \right)U_{n\; f}\frac{\partial\theta_{n\; f}}{\partial X}} = {\frac{\partial}{\partial Y}\left( {\left( {\frac{k_{o}}{k_{f}} + {\lambda\frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\; U_{n\; f}}} \right)\frac{\partial\theta_{n\; f}}{\partial Y}} \right)}} & {{Eq}.\mspace{14mu} 174}\end{matrix}$while the energy equation for the volume containing the pure fluid is:

$\begin{matrix}{{\left( P_{e} \right)_{f}U_{f}\frac{\partial\theta_{f}}{\partial X}} = \frac{\partial^{2}\theta_{f}}{\partial Y^{2}}} & {{Eq}.\mspace{14mu} 175}\end{matrix}$

Different distributions for the nanoparticles of the dispersive elementswill be analyzed in this work. In one of these distributions, the regionthat is active with thermal dispersion effects is considered to be arectangular region of height 2l around the channel's centerline as shownin FIG. 74A. Another distribution considers the region comprisingthermal dispersion effects to be present only at the two identicalrectangular regions of height l attached to the channel's plates asshown in FIG. 74B.

The boundary conditions for the central arrangement are

$\begin{matrix}{\frac{\mathbb{d}{\theta_{n\; f}\left( {X,0} \right)}}{\mathbb{d}Y} = 0} & {{{Eq}.\mspace{14mu} 176}a} \\{{{\left( {\frac{k_{o}}{k_{f}} + {\lambda\frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\;{U(\Lambda)}}} \right)\frac{\mathbb{d}{\theta_{n\; f}\left( {X,\Lambda} \right)}}{\mathbb{d}Y}} = \frac{\mathbb{d}{\theta_{f}\left( {X,\Lambda} \right)}}{\mathbb{d}Y}},} & {{{Eq}.\mspace{14mu} 176}b} \\{{\theta_{f}\left( {X,\Lambda} \right)} = {\theta_{n\; f}\left( {X,\Lambda} \right)}} & {{{Eq}.\mspace{14mu} 176}c} \\{\frac{\mathbb{d}{\theta_{f}\left( {X,1} \right)}}{\mathbb{d}Y} = 1} & {{{Eq}.\mspace{14mu} 176}d}\end{matrix}$while the boundary conditions for the second arrangement (boundaryarrangement) are

$\begin{matrix}{\mspace{79mu}{\frac{\mathbb{d}{\theta_{f}\left( {X,0} \right)}}{\mathbb{d}Y} = 0}} & {{{Eq}.\mspace{14mu} 177}a} \\{{{\left( {\frac{k_{o}}{k_{f}} + {\lambda\frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\;{U\left( {1 - \Lambda} \right)}}} \right)\frac{\mathbb{d}{\theta_{n\; f}\left( {X,{1 - \Lambda}} \right)}}{\mathbb{d}Y}} = \frac{\mathbb{d}{\theta_{f}\left( {X,{1 - \Lambda}} \right)}}{\mathbb{d}Y}},} & {{{Eq}.\mspace{14mu} 177}b} \\{\mspace{79mu}{{\theta_{f}\left( {X,{1 - \Lambda}} \right)} = {\theta_{n\; f}\left( {X,{1 - \Lambda}} \right)}}} & {{{Eq}.\mspace{14mu} 177}c} \\{\mspace{79mu}{{\left( {\frac{k_{o}}{k_{f}} + {\lambda\frac{\left( {\rho\; c_{p}} \right)_{{n\; f}\;}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\;{U(1)}}} \right)\frac{\mathbb{d}{\theta_{f}\left( {X,1} \right)}}{\mathbb{d}Y}} = 1}} & {{{Eq}.\mspace{14mu} 177}d}\end{matrix}$where Λ=l/h. Other distributions for the dispersive elements will beconsidered later such as the parabolic distribution and the exponentialdistribution.

For thermal fully developed conditions, axial gradient of thetemperature reaches a constant value equal to dT/dx. That is, the heatflux is equal to:

$\begin{matrix}{q = {\frac{\mathbb{d}T}{\mathbb{d}x}\left( {{\left( {1 - \varphi_{c\; f}} \right)\left( {\rho\; c_{p}} \right)_{f}\left( u_{m} \right)_{f}} + {{\varphi_{c\; f}\left( {\rho\; c_{p}} \right)}_{n\; f}\left( u_{m} \right)_{n\; f}}} \right)h}} & {{Eq}.\mspace{14mu} 178}\end{matrix}$where φ_(cf) is the ratio of the volume comprising thermal dispersioneffects to the total channel volume. (u_(m))_(f) is the average velocityin the fluid phase while (u_(m))_(nf) is the average velocity in thenanofluid or the region containing the thermal dispersive elements.

As such, Equation 174 and Equation 175 reduce to

$\begin{matrix}{{A\; U_{n\; f}} = {\frac{\partial}{\partial Y}\left( {\left( {K + {E\;\varphi\; U}} \right)\frac{\partial\theta_{n\; f}}{\partial Y}} \right)}} & {{Eq}.\mspace{14mu} 179} \\{{G\; U_{f}} = \frac{\partial^{2}\theta_{f}}{\partial Y^{2}}} & {{Eq}.\mspace{14mu} 180} \\{{{{where}\mspace{14mu} A} = {P\;{e_{f}\left( \frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}} \right)}\frac{\mathbb{d}\theta_{n\; f}}{\mathbb{d}X}}},{K = {k_{o}/k_{f}}},} & \; \\{E = {{\lambda\frac{\left( {\rho\; c_{p}} \right)_{n\; f}}{\left( {\rho\; c_{p}} \right)_{f}}\varphi\mspace{14mu}{and}\mspace{14mu} G} = {\left( P_{e} \right)_{f}{\frac{d\;\theta_{f}}{\partial X}.}}}} & \;\end{matrix}$

Since (ρc_(p))_(nf) does not vary significantly when the volume fractionof the ultrafine particles or the dispersive elements is less than 4% asused in the literature, A and G are almost equal to unity

$\left( {{{Pe}_{f}\frac{\mathbb{d}\theta}{\mathbb{d}X}} = {{\left( \frac{\rho\; c_{p}u_{m}h}{k} \right)\frac{\mathbb{d}\left( {T\;{k/\left( {q\; h} \right)}} \right)}{\mathbb{d}\left( {x/h} \right)}} = {{\frac{\rho\; c_{p}u_{m}h}{q}\frac{\mathbb{d}T}{\mathbb{d}x}} = 1.0}}} \right).$8B. Analytical Solutions

Consider a uniform flow inside the channel such that U=1. Equation 179and Equation 180 reduce to

$\begin{matrix}{\frac{\partial^{2}\theta_{n\; f}}{\partial Y^{2}} = \frac{1}{\left( {K + E} \right)}} & {{{Eq}.\mspace{14mu} 181}a} \\{\frac{\partial^{2}\theta_{f}}{\partial Y^{2}} = 1} & {{{Eq}.\mspace{14mu} 181}b}\end{matrix}$

The solution to Equation 181a and Equation 181b for the centralarrangement of the dispersive elements is Equation 182a

${\frac{{\theta_{W}(X)} - {\theta_{n\; f}\left( {X,Y} \right)}}{{\theta_{W}(X)} - {\theta_{m}(X)}} \cong \frac{{1.5\left( {\Lambda^{2} - Y^{2}} \right)} + {1.5\left( {K + E} \right)\left( {1 - \Lambda^{2}} \right)}}{\Lambda^{3} - {\left( {K + E} \right)\left( {\Lambda^{3} - {1.5\Lambda^{2}} + 0.5} \right)} + \mspace{79mu}{1.5\left( {K + E} \right)\left( {1 - \Lambda^{2}} \right)}}},{0 < Y < \Lambda}$and Equation 182b

${\frac{{\theta_{W}(X)}\; - \;{\theta_{f}\left( {X,\; Y} \right)}}{{\theta_{W}(X)}\; - \;{\theta_{m}(X)}} \cong \frac{{1.5\;\left( {1 - Y^{2}} \right)\left( {K + E} \right)}\;}{\Lambda^{3} - {\left( {K + E} \right)\;\left( {\Lambda^{3} - {1.5\;\Lambda^{2}} + 0.5} \right)}\; + \mspace{79mu}{1.5\;\left( {K + E} \right)\;\left( {1 - \Lambda^{2}} \right)}}},{\Lambda < Y < 1}$where θ_(W) is the plate temperature at a given section X. The parameterθ_(m) is the mean bulk temperature. It is defined as

$\begin{matrix}{{\theta_{m}(X)} = {\int_{0}^{1}{{U(Y)}{\theta\left( {X,Y} \right)}{\mathbb{d}Y}}}} & {{Eq}.\mspace{14mu} 183}\end{matrix}$

As such, the fully developed value for the Nusselt number is

$\begin{matrix}\begin{matrix}{{Nu}_{f\; d} = {\frac{h_{c}h}{k_{f}} = {\frac{1}{{\theta_{W}(X)} - {\theta_{m}(X)}} = \frac{1}{{\theta_{f}\left( {X,1} \right)} - {\theta_{m}(X)}}}}} \\{\cong \frac{3\left( {K + E} \right)}{\Lambda^{3} - {\left( {K + E} \right)\left( {\Lambda^{3} - {1.5\;\Lambda^{2}} + 0.5} \right)} + {1.5\left( {K + E} \right)\left( {1 - \Lambda^{2}} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} 184}\end{matrix}$where h_(c) is the convective heat transfer coefficient at the channel'splate.

For the second type of arrangements for the thermal dispersion region.The solution for Equation 181a and Equation 181b is Equation 185a

${\frac{{\theta_{W}(X)} - {\theta_{f}\left( {X,Y} \right)}}{{\theta_{W}(X)} - {\theta_{m}(X)}} \cong \frac{1.5\left( {1 + {\left( {1 - \Lambda^{2}} \right)\left( {K + E} \right)\mspace{14mu}\left( {1 - Y^{2}} \right)} - \left( {1 - \Lambda} \right)^{2}} \right)}{K - {\left( {K - 1} \right)\left( {\Lambda^{3} - {3\;\Lambda^{2}} + {3\;\Lambda}} \right)} + {E\left( {1 - \Lambda} \right)}^{3}}},{0 < Y < \Lambda}$and Equation 185b

${\frac{{\theta_{W}(X)} - {\theta_{{n\; f}\;}\left( {X,Y} \right)}}{{\theta_{W}(X)} - {\theta_{m}(X)}} \cong \frac{1.5\left( {1 - Y^{2}} \right)}{K - {\left( {K - 1} \right)\left( {\Lambda^{3} - {3\;\Lambda^{2}} + {3\;\Lambda}} \right)} + \mspace{115mu}{E\left( {1 - \Lambda} \right)}^{3}}},{\Lambda < Y < 1}$

The corresponding fully developed value for Nusselt number for this caseis

$\begin{matrix}\begin{matrix}{{Nu}_{f\; d} = {\frac{h_{c}h}{k_{f}} = {\frac{1}{{\theta_{W}(X)} - {\theta_{m}(X)}} = \frac{1}{{\theta_{n\; f}\left( {X,1} \right)} - {\theta_{m}(X)}}}}} \\{\cong \frac{3\left( {K + E} \right)}{K - {\left( {K - 1} \right)\left( {\Lambda^{3} - {3\;\Lambda^{2}} + {3\;\Lambda}} \right)} + {E\left( {1 - \Lambda} \right)}^{3}}}\end{matrix} & {{Eq}.\mspace{14mu} 186}\end{matrix}$8C. Volume Fraction of the Dispersive Elements

The total number of dispersive elements is considered to be fixed foreach distribution. As such, the volume fraction of the dispersiveelement for the central or the boundary arrangements is related to theirthickness according to the following relation:

$\begin{matrix}{\varphi = {\frac{\varphi_{o}h}{l} = \frac{\varphi_{o}}{\Lambda}}} & {{Eq}.\mspace{14mu} 187}\end{matrix}$where φ_(o) is the volume fraction of the dispersive elements when theyare uniformly filling the whole channel volume. Utilizing Equation 187,the parameter E utilized in Equation 179 and Equation 180 can beexpressed according to the following:

$\begin{matrix}{E = {{E_{o}\left( \frac{h}{l} \right)} = \frac{E_{o}}{\Lambda}}} & {{Eq}.\mspace{14mu} 188}\end{matrix}$where E_(o) is named as the thermal dispersion parameter.8D. Other Spatial Distribution for the Dispersive Elements

Practically, it is difficult to have the dispersive elementsconcentrated in a region while the other region is a pure fluid. Assuch, two other distributions for the dispersive elements are consideredin this work. They are the exponential and the parabolic distributionsas illustrated in the following:

$\begin{matrix}{\varphi = {\varphi_{o}\left( {1 + {D_{c}\left( {\frac{1}{3} - \left( \frac{y}{h} \right)^{2}} \right)}} \right)}} & {{Eq}.\mspace{14mu} 189} \\{\varphi = {\frac{\varphi_{o}D_{e}}{{\mathbb{e}}^{D_{e}} - 1}{\mathbb{e}}^{D_{e}Y}}} & {{Eq}.\mspace{14mu} 190}\end{matrix}$

Note that the average volume fraction for each distribution is φ_(o)irrespective to values of D_(e) and D_(p). One of the objectives of ourwork is to obtain the values of D_(c) and D_(e) and Λ that producesmaximum heat transfer inside the channel.

The excess in Nusselt number κ is defined as the ratio of the maximumNusselt number that can be obtained by having a certain volume fractiondistribution (Nu_(nd)) to the Nusselt number corresponding to a uniformdistribution of the dispersive elements (Nu_(ud)). It is expressed asfollows:

$\begin{matrix}{\kappa = \frac{{Nu}_{n\; d}}{{Nu}_{u\; d}}} & {{Eq}.\mspace{14mu} 191}\end{matrix}$

It can be shown that Equation 191 exhibits a local maximum or minimumvalue at specific thermal dispersion parameter (E_(o)*)_(critical) forthe boundary arrangement. This is related to the dimensionless thicknessof the dispersive region through the following relation:

$\begin{matrix}{\frac{\left( E_{o}^{*} \right)_{critical}}{K\;\Lambda} = {{- 1} + \sqrt{\frac{\left( {{K\left( {\Lambda^{3} - {3\;\Lambda^{2}} + 3} \right)} + {\left( {1 - \Lambda} \right)\left( {\Lambda - 2} \right)}} \right)}{\left( {1 - \Lambda} \right)^{3}}}}} & {{Eq}.\mspace{14mu} 192}\end{matrix}$8E. Numerical Methods

Equation 174 and Equation 175 were descritized using three pointscenteral differencing in the Y direction while backward differencing wasutilized for the temperature gradient in the X-direction. The resultingtri-diagonal system of algebraic equations at X=ΔX was then solved usingthe well established Thomas algorithm. See Blottner, F. G. (1970) AIAAJ. 8:193-205. The same procedure was repeated for the consecutiveX-values until X reached the value of B/h. Equation 179 and Equation 180were also descritized using three points central differencing and solvedusing Thomas algorithm.

8F. Thermal Dispersion Effects for the Central and Boundary Arrangements

FIG. 75 shows the variation of the fully developed Nusselt number withthe thermal dispersion parameter E_(o) and the dimensionless thicknessof the thermally dispersed region Λ for the central arrangement. Forlower values of Λ, the Nusselt number does not change due toconcentrations of the thermal dispersive elements around the center ofthe channel. However, as the thickness of the dispersive regionincreases, it will have a profound effect on the Nusselt number. Themotion of nanoparticles or the dispersive elements within the core flowof the channel produces a negligible change in the heat transfercharacteristics as shown in FIG. 75. The Nusselt number increases as Λincreases to a maximum value and then starts to decrease when thedispersive elements are concentrated according to the boundaryarrangement. See FIG. 76. The arrangement shown in FIG. 76 illustratesthat a specific distribution for the same dispersive elements canenhance the heat transfer. This distribution is a function of E_(o) andthe velocity profile as shown in FIG. 76. In this figure, the thermaldispersive region thickness Λ that produces the optimum enhancement inthe Nusselt number is shown to increase as the E_(o) increases. As such,flow and thermal conditions along with the properties of the dispersiveelements such as their sizes and their surface geometry determine thedistribution of the dispersive elements that result in a maximumenhancement in the heat transfer.

8G. Thermal Dispersion Effects for the Central and Boundary Arrangementsat Thermally Developing Conditions

FIG. 77 illustrates the effects of the dispersion coefficient C* on theNusselt number at the exit for various thicknesses of the thermallydispersed region A arranged with the central configuration. These valuesare for a thermally developing condition as the minimum Nusselt numberin this figure is greater than the corresponding value at thermallydeveloped conditions illustrated in FIG. 75. This figure shows that whenΛ is below 0.35, heat transfer is almost unaffected by thermaldispersion. As can be seen, the average plate temperature shown in FIG.78 (Pe_(f)=670) is almost unchanged when Λ is below 0.37 while it isbelow 0.5 in FIG. 79 (Pe_(f)=1340) for the central arrangement.Similarly, the maximum Nusselt number or the minimum average platetemperatures at lower Pe_(f) values occur at higher values of Λ comparedto those at higher Pe_(f) values for different boundary arrangements ascan be noticed from FIG. 76, FIG. 80, FIG. 81 and FIG. 82. This isbecause temperature gradients near the core flow increase as Pe_(f)decreases thus thermal dispersion effects are increased.

8G. Thermal Dispersion Effects on the Excess In The Nusselt Number atThermally Fully Developed Conditions

FIG. 83 and FIG. 84 illustrate various proposed volume fractiondistributions for the same nanoparticles. As shown in FIG. 85, theNusselt number reaches a maximum value when E_(o)>0 for the exponentialdistribution of the dispersive elements while the parabolic distributionproduces no maxima in the Nusselt number. The excess in Nusselt number κis always greater than one for the boundary arrangement while it isgreater than one for the exponential distribution when the velocity isuniform as shown in FIG. 87. The excess in Nusselt number increases asE_(o) increases and reaches a constant value equal to 1.12 for theparabolic velocity profile along with the boundary arrangement for thedispersive elements while it is 1.21 for the uniform velocity profile.This indicates that almost 12% increase in the heat transfer can beachieved in highly dispersive media when the dispersive elements areconcentrated near the boundary for the parabolic velocity profile. Theexponential distribution produced a maximum excess in the Nusselt numberequal to 1.18 for uniform velocity profile. The latter results can beused to model Darcian flow inside a channel filled with a porous mediumhaving a uniform porosity and comprising dispersive elementsexponentially distributed along the center line of the channel. Thesefigures illustrate the importance of flow conditions and thedistribution of the dispersive elements on the degree of enhancement inheat transfer.

8. CONCLUSION

Enhancements in heat transfer are investigated inside channels filledwith a fluid having different thermal dispersive properties. Differentdistributions for dispersive elements such as nanoparticles or flexiblehairy tubes extending from the channel plates are considered. Thedispersive elements are considered to be uniformly distributed in thecentral region, near the boundaries, having an exponential distributionand having a parabolic distribution.

The boundary arrangement and the exponential distribution of thedispersive elements were shown to produce substantial enhancements inheat transfer compared to the case when the dispersive elements areuniformly distributed. The presence of the dispersive elements in coreregion produced no significant change in the heat transfer. The maximumexcess in Nusselt number was found to be 1.21 using the boundaryarrangement for the volume fraction with uniform flow while theparabolic velocity profile produced a maximum excess in Nusselt numberequal to 1.12. The volume fraction distribution that maximizes the heattransfer is governed by the flow and thermal conditions as well as theproperties of dispersive elements. This work demonstrates that supernanofluids or super dispersive media can be prepared by controlling thethermal dispersion properties inside the fluid.

To the extent necessary to understand or complete the disclosure of thepresent invention, all publications, patents, and patent applicationsmentioned herein are expressly incorporated by reference therein to thesame extent as though each were individually so incorporated.

Having thus described exemplary embodiments of the present invention, itshould be noted by those skilled in the art that the within disclosuresare exemplary only and that various other alternatives, adaptations, andmodifications may be made within the scope of the present invention.Accordingly, the present invention is not limited to the specificembodiments as illustrated herein, but is only limited by the followingclaims.

1. A device comprising at least one microchannel defined by at least oneflexible seal, at least one flexible complex seal, or a combinationthereof, and at least one immobile and inflexible substrate and at leastone mobile and inflexible substrate, wherein said at least onemicrochannel comprises a plurality of double layered microchannels. 2.The device of claim 1, wherein the mobile and inflexible substrate iscapable of movement in a normal direction due to an expansion or acontraction of said at least one flexible seal or said at least oneflexible complex seal.
 3. The device of claim 1, wherein said at leastone flexible seal separates said at least one and inflexible substratefrom said at least one mobile and inflexible substrate.
 4. The device ofclaim 1, wherein the flexible seal connects the immobile and inflexiblesubstrate and the mobile and flexible substrate.
 5. The device of claim1, wherein at least one of said at least one immobile and inflexiblesubstrate and said at least one mobile and inflexible substrate isheated.
 6. The device of claim 1, and further comprising at least oneheated substrate.
 7. The device of claim 1, wherein the distance betweenthe substrates increases when the average pressure between thesubstrates increases.
 8. A device comprising at least one microchanneldefined by at least one flexible seal, at least one flexible complexseal, or a combination thereof, and at least one immobile and inflexiblesubstrate and at least one mobile and inflexible substrate, wherein saidat least one immobile and inflexible substrate and said at least onemobile and inflexible substrate separate a plurality of fluid layers. 9.The device of claim 8, wherein the direction of the fluid flow of thefluid layers is the same.
 10. The device of claim 8, wherein the rate ofthe fluid flow of the fluid layers is the same.
 11. The device of claim8, wherein the direction of the fluid flow of the fluid layers isdifferent.
 12. The device of claim 8, wherein the rate of the fluid flowof the fluid layers is different.
 13. A device comprising at least onemicrochannel defined by at least one flexible seal, at least oneflexible complex seal, or a combination thereof, and at least oneimmobile and inflexible substrate and at least one mobile and inflexiblesubstrate, wherein a liquid coolant with super dispersive media flows inthe volumetric space of the at least one microchannel.
 14. The device ofclaim 13, wherein the super dispersive media comprises contains at leastone metallic nanoparticle, at least one carbon nanoparticle, at leastone nanotube, and at least one flexible nanostring.
 15. The device ofclaim 14, wherein the super dispersive media is non-uniformlydistributed in the volumetric space of the at least one microchannel.16. The device of claim 15, wherein the distribution of the superdispersive media is minimum in the regions with least transverseconvection heat transfer in the volumetric space of the microchannel.17. The device of claim 15, wherein the distribution of the superdispersive media is maximum in the regions with maximum transverseconvection heat transfer in the volumetric space of the microchannel.